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fractional Legendre

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A research paper by Yajun Zhou (arXiv 1301.1735v4, April 2013) that derives a closed form for the integral of [P_ν(x)]^2 P_ν(-x) over -1 to 1 and evaluates the finite Hilbert transform of P_ν(x)P_ν(-x). It proves J. G. Wan's conjectured identity for integrals of complete elliptic integrals K, involving Γ(1/4)^8/(128π²). It also reviews fractional-degree Legendre functions P_{-1/2}, P_{-1/3}, P_{-1/4}, P_{-1/6} and their link to K. It sits in a support folder for Phil's toroidal-coordinates electrostatics work.

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arXiv:1301.1735v4 [math.CA] 29 Apr 2013Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals Yajun Zhou PROGRAM IN APPLIED AND COMPUTATIONAL MATHEMATICS , PRINCETON UNIVERSITY , PRINCETON , NJ 08544 A closed-form formula is derived for the generalized Clebsc h-Gordan integral/integraltext 1 −1[Pν(x)]2Pν(−x)dx, with Pνbeing the Legendre function of arbitrary complex degree ν∈C. The ﬁnite Hilbert transform of Pν(x)Pν(−x),−1<x<1 is evalu- ated. An analytic proof is provided for a recently conjectur ed identity/integraltext 1 0[K(/radicalbig 1−k2)]3dk=6/integraltext 1 0[K(k)]2K(/radicalbig 1−k2)kdk= [Γ(1 4)]8/(128 π2) involving complete elliptic integrals of the ﬁrst kind K(k) and Euler’s gamma function Γ(z). 0 Introduction In a wide range of mathematical and physical contexts, one ne eds to evaluate multiple elliptic integrals where the integrands involve one or more factors of complete ellip tic integrals. As a glimpse of some recent applications in number theory, high-energy physics, statistical mechanic s and probability theory, we mention automorphic Green’s functions [1], multi-loop Feynman diagrams [2], lattice Gr een’s functions [3] and short uniform random walks [4, 5, 6]. Various techniques, of algebraic, geometric, combinat oric or analytic ﬂavor, have been developed to express these multiple elliptic integrals in terms of familiar mathemati cal constants and special values of Euler’s gamma function [1, 2, 3, 4, 5, 6]. When the integrands contain the products of three or more com plete elliptic integrals, the evaluation can become analytically challenging. For example, the following rela tion [Γ(1 4)]8 128π2?=/integraldisplay 1 0/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig3 dk?=6/integraldisplay 1 0[K(k)]2K/parenleftBig/radicalbig 1−k2/parenrightBig kdk has been recently conjectured by J. G. Wan in [6] based on nume rical evidence. In this work, we solve Wan’s conjecture and compute a related family of multiple elliptic integrals with sophisticated appearance. We employ an approach that is probably overlook ed in previous literature: connecting elliptic integrals to spherical functions and spherical rotations. Concretel y speaking, the complete elliptic integral of the ﬁrst kind K(k) :=/integraltext π/2 0(1−k2sin2θ)−1/2dθis related to fractional degree Legendre functions P−1/2,P−1/3,P−1/4,P−1/6, where Pν(1)=1;Pν(cosθ) :=2 π/integraldisplay θ 0cos(2ν+1)β 2/radicalbig 2(cos β−cosθ)dβ,θ∈(0,π),ν∈C extend the familiar Legendre polynomials Pℓ(x)=1 2ℓℓ!dℓ dxℓ[(x2−1)ℓ],ℓ∈Z≥0 to arbitrary complex degree ν∈C. By a modest generalization of the “spherical harmonic coup ling” for Legendre poly- nomials (Clebsch-Gordan theory) to Legendre functions of a rbitrary degree, we are able to prove the integral identity /integraldisplay 1 −1[Pν(x)]2Pν(−x)dx=limz→ν1+2cos( πz) 3πΓ/parenleftbigz+1 2/parenrightbig Γ/parenleftbig3z+2 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig3Γ/parenleftbig3z+3 2/parenrightbig,ν∈C and compute the ﬁnite Hilbert transform P/integraldisplay 1 −12Pν(ξ)Pν(−ξ) π(x−ξ)dξ=[Pν(x)]2−[Pν(−x)]2 sin(νπ),ν∈C/integerdivideZ. 1 The pair of formulae above, along with a couple of geometric t ransformations related to rotations on spheres, allow us to evaluate a slew of multiple elliptic integrals, which emb ody Wan’s conjectural identity as a special case. The article is organized as follows. In §1, we review some cla ssical results that express fractional degree Legendre functions as complete elliptic integrals, and give a new pro of of Hobson’s coupling formula for two Legendre functions, based on scaling limits. We carry on the scaling analysis in § 2, so as to interpolate the standard Clebsch-Gordan in- tegrals for Legendre polynomials into a closed-form expres sion for/integraltext 1 −1[Pν(x)]2Pν(−x)dxwith arbitrary complex degree ν∈C. This is followed by a geometric intermezzo on spherical rot ations in §3, providing tools to transform one multiple elliptic integral into another, as well as preparing some in tegral identities for §4. The ﬁnite Hilbert transform (also known as the Tricomi transform) of the function Pν(x)Pν(−x) is studied in §4, in the spirit of “spherical harmonic cou- pling”. Lastly, §5 serves as a perspective on subsequent wor ks on multiple elliptic integrals related to, or motivated b y Legendre functions. 1 Fractional Degree Legendre Functions and Hansen-Heine Sc aling Limits The Legendre functions of the ﬁrst kind can be deﬁned via the M ehler-Dirichlet integral Pν(1)=1;Pν(cosθ) :=2 π/integraldisplay θ 0cos(2ν+1)β 2/radicalbig 2(cos β−cosθ)dβ,θ∈(0,π),ν∈C. (1) Alternatively, we may characterize Pν(x),−1<x≤1 as the unique C2(−1,1) solution to d dx/bracketleftbigg (1−x2)df(x) dx/bracketrightbigg +ν(ν+1)f(x)=0,f(1)=1. (2) By either deﬁnition, we have Pν(x)≡P−ν−1(x). Later in this work, we shall also use the Legendre function s of the second kind, given by Qν(x) :=π 2sin( νπ)[cos(νπ)Pν(x)−Pν(−x)],ν∈C/integerdivideZ;Qn(x) :=lim ν→nQν(x),n∈Z≥0 (3) for−1<x<1. By convention, the Legendre function of the second kind Qn(x) is undeﬁned for any negative integer degree n∈Z<0. While both Pν(x) and Qν(x) solve the same Legendre differential equation of degree ν∈C/integerdivideZ<0, the function Qν(x) blows up logarithmically as x→1−0+, unlike the natural boundary condition for Pν(1)≡1. Additionally, the Legendre function of the ﬁrst kind can be r epresented as the Laplace integral under some restric- tions: Pν(z) :=1 2π/integraldisplay 2π 0eνlog(z+i/radicallow 1−z2cosφ)dφ≡1 2π/integraldisplay 2π 0eνlog(z−i/radicallow 1−z2cosφ)dφ,Rez>0,ν∈C. (4) 1.1 Kleiber-Ramanujan Correspondence Highlights of the relation P−1/2(cosθ)=(2/π)K(sin(θ/2)) date back to J. Kleiber’s posthumous publication [7] in 1893, where systematic studies were carried out on connections be tween complete elliptic integrals and Legendre functions P(2n+1)/2,n∈Z, the latter of which are useful for solving differential equ ations in toroidal coordinates [Ref. 8, §§253-258]. In his seminal paper [9] and legendary notebooks [Ref. 10, Ch ap. 33], S. Ramanujan developed rich and elegant theories relating P−1/3,P−1/4,P−1/6to complete elliptic integrals of the ﬁrst kind. We recapitulate the aforementioned classical results in th e following short lemma. Lemma 1.1 (Some Fractional Degree Legendre Functions) (a)We have the following identities for 0≤θ<π: P−1/2(cosθ)=2 πK/parenleftbigg sinθ 2/parenrightbigg =2 π/integraldisplay π/2 0dψ/radicalbig 1−sin2(θ/2)sin2ψ, (5) P−1/4(cosθ)=2 π1/radicallow 1+sin(θ/2)K/parenleftBigg/radicalBigg 2sin( θ/2) 1+sin(θ/2)/parenrightBigg =2 π/integraldisplay π/2 0dψ/radicalbig 1−sin2(θ/2)sin4ψ, (6) 2 (b)For a parameter p∈[0,1), the following relations P−1/6/parenleftbigg 1−227p2(1+p)2 4(1+p+p2)3/parenrightbigg =/radicalBigg 1+p+p2 1+2pP−1/2/parenleftbigg 1−2p(2+p) 1+2p/parenrightbigg =2 π/radicalBigg 1+p+p2 1+2pK/parenleftBigg/radicalBigg p(2+p) 1+2p/parenrightBigg , (7a) P−1/6/parenleftbigg 227p2(1+p)2 4(1+p+p2)3−1/parenrightbigg =/radicalBigg 1+p+p2 1+2pP−1/2/parenleftbigg 2p(2+p) 1+2p−1/parenrightbigg =2 π/radicalBigg 1+p+p2 1+2pK/parenleftBigg/radicalBigg 1−p2 1+2p/parenrightBigg ; (7b) hold for degree ν=−1/6, and the transformations P−1/3/parenleftbigg 1−227p2(1+p)2 4(1+p+p2)3/parenrightbigg =1+p+p2 /radicalbig 1+2pP−1/2/parenleftbigg 1−2p3(2+p) 1+2p/parenrightbigg =2 π1+p+p2 /radicalbig 1+2pK/parenleftBigg/radicalBigg p3(2+p) 1+2p/parenrightBigg , (8a) P−1/3/parenleftbigg 227p2(1+p)2 4(1+p+p2)3−1/parenrightbigg =1+p+p2 /radicalbig 3+6pP−1/2/parenleftbigg 2p3(2+p) 1+2p−1/parenrightbigg =2 π1+p+p2 /radicalbig 3+6pK/parenleftBigg/radicalBigg (1−p)(1+p)3 1+2p/parenrightBigg . (8b) apply to degree ν=−1/3. Proof (a) Let E(k)=/integraltext π/2 0/radicalbig 1−k2sin2φdφbe the complete elliptic integral of the second kind, then we have the differ- ential relations dK(k) dk=E(k) k(1−k2)−K(k) k,dE(k) dk=E(k)−K(k) k. It is thus routine to check that the proposed forms of ellipti c integrals in Eqs. 5 and 6 satisfy the said Legendre differential equations and natural boundary conditions (E q. 2). The last integral in Eq. 5 is the standard deﬁnition for the co mplete elliptic integral of the ﬁrst kind. With a variable substitution ϑ=arctan(/radicallow 1+sin(θ/2)tan ψ) [Ref. 11, p. 105], one can convert the last expression in Eq. 6 into the standard form. (b) As in the proof of part (a), one can verify the relations be tween the respective fractional degree Legendre functions and complete elliptic integrals by a routine, though time-c onsuming procedure of Legendre differential equations. Equations 7a and 7b have appeared in Ramanujan’s work [Ref. 1 0, pp. 161-163] on the elliptic function theory for sig- nature 6. As suggested by [10], Eqs. 7a, 7b can be constructed from ﬁrst principles, using a hypergeometric transforma- tion that identiﬁes P−1/6(cosθ)=2F1/parenleftbig1/6,5/6 1/vextendsingle/vextendsinglesin2θ 2/parenrightbig ,0≤θ≤π/2 with the Fricke-Klein function 2F1/parenleftbig1/12,5/12 1/vextendsingle/vextendsinglesin2θ/parenrightbig ,0≤ θ≤π/2 [Ref. 12, p. 334], the latter of which has a well-known conn ection to complete elliptic integrals of the ﬁrst kind [1]. We may combine Eqs. 7a and 7b into a single formula P−1/6/parenleftbiggx(9−x2) (3+x2)3/2/parenrightbigg =/radicalBigg/radicallow 3+x2 2P−1/2(x)=2 π/radicalBigg/radicallow 3+x2 2K/parenleftBigg/radicalbigg 1−x 2/parenrightBigg ,x∈(−1,1], (9) a form that is handier for conversion between integrals invo lving P−1/6andK. Ramanujan’s studies of the elliptic function theory for sig nature 3 bring us Eqs. 8a and 8b. One may consult [Ref. 10, pp. 112-114] for the detailed theoretical arguments lea ding to the discovery of the algebraic relations between P−1/3 andK. /squaresolid According to the recursion relation for Legendre functions (x2−1)dPν(x)/dx=νxPν(x)−νPν−1(x) and the symmetry Pν(x)=P−ν−1(x), one can relate some fractional degree Legendre functions to complete elliptic integrals of the ﬁrst and the second kinds, such as P1/2(cosθ)=2 π/bracketleftbigg 2E/parenleftbigg sinθ 2/parenrightbigg −K/parenleftbigg sinθ 2/parenrightbigg/bracketrightbigg ,P3/2(cosθ)=2 3π/bracketleftbigg 8cosθE/parenleftbigg sinθ 2/parenrightbigg −(4cos θ+1)K/parenleftbigg sinθ 2/parenrightbigg/bracketrightbigg , P1/4(cosθ)=2 π/bracketleftBigg 2/radicallow 1+sin(θ/2)E/parenleftBigg/radicalBigg 2sin( θ/2) 1+sin(θ/2)/parenrightBigg −1/radicallow 1+sin(θ/2)K/parenleftBigg/radicalBigg 2sin( θ/2) 1+sin(θ/2)/parenrightBigg/bracketrightBigg , P3/4(cosθ)=2 3π/bracketleftBigg 2/radicallow 1+sin(θ/2)E/parenleftBigg/radicalBigg 2sin( θ/2) 1+sin(θ/2)/parenrightBigg −1−2cosθ/radicallow 1+sin(θ/2)K/parenleftBigg/radicalBigg 2sin( θ/2) 1+sin(θ/2)/parenrightBigg/bracketrightBigg . 3 1.2 Hansen-Heine Scaling Limits and Hobson Coupling Formul a By taking scaling limits of certain identities for Legendre polynomials, one may extend their validity to Legendre functions of arbitrary complex degree. In this manner, one c an compute some multiple elliptic integrals by interpolati ng Legendre polynomial identities, the latter of which often h ave clear motivations from the harmonic analysis on spheres . As we may recall, for positive integers m, the associated Legendre functions of degree νand order mare deﬁned by Pm ν(x) :=(−1)m(1−x2)m/2dmPν(x) dxm, P−m ν(x) :=(−1)mΓ(ν−m+1) Γ(ν+m+1)Pm ν(x); Qm ν(z) :=(z2−1)m/2dmQν(z) dzm, Q−m ν(z) :=Γ(ν−m+1) Γ(ν+m+1)Qm ν(x) One also writes P0 ν=PνandQ0 ν=Qν. The spherical harmonics Yℓm(θ,φ) :=/radicalBigg 2ℓ+1 4π(ℓ−m)! (ℓ+m)!Pm ℓ(cosθ)eimφ,ℓ∈Z≥0;m∈Z∩[−ℓ,ℓ] carry the Condon-Shortley phase Yℓm(θ,φ)=(−1)mYℓ,−m(θ,φ). In the following, we show that the studies of Legendre functi ons of arbitrary degree can be built on the Mehler- Dirichlet formula for spherical harmonics (see item 8.927 i n [13] or §4.5.4 in [14]), which couples two normal vectors n1(θ1,φ1)=(sinθ1cosφ1,sinθ1sinφ1,cosθ1) and n2(θ2,φ2)=(sinθ2cosφ2,sinθ2sinφ2,cosθ2) on the unit sphere: 4π∞/summationdisplay ℓ=0ℓ/summationdisplay m=−ℓYℓm(θ1,φ1)Yℓm(θ2,φ2)cos[( ℓ+1 2)β] 2ℓ+1=∞/summationdisplay ℓ=0Pℓ(n1·n2)cos[( ℓ+1 2)β]=θH(cosβ−n1·n2)/radicalbig 2(cos β−n1·n2),|n1|=|n2|=1.(10) Here, the Heaviside theta function is deﬁned as θH(x)=(1+x |x|)/2,x∈R/integerdivide{0}, and the range of the angle βis (0,π). Lemma 1.2 (Hobson Coupling Formula) Forθ1,θ2∈[0,π),ν∈C, the Legendre function of the ﬁrst kind satisﬁes the Hobson coupling formula 1 2π/integraldisplay 2π 0Pν(cosθ1cosθ2+sinθ1sinθ2cosφ)dφ=/braceleftBigg Pν(cosθ1)Pν(cosθ2), θ1+θ2≤π Pν(−cosθ1)Pν(−cosθ2),θ1+θ2≥π(11) which specializes to the following integral identities for θ1,θ2∈[0,π): 1 4/integraldisplay 2π 0K/parenleftbigg sinΘ 2/parenrightbigg dφ=  K/parenleftBig sinθ1 2/parenrightBig K/parenleftBig sinθ2 2/parenrightBig ,θ1+θ2≤π K/parenleftBig cosθ1 2/parenrightBig K/parenleftBig cosθ2 2/parenrightBig ,θ1+θ2≥π(12) 1 4/integraldisplay 2π 0K/parenleftBigg/radicalBigg 2sin( Θ/2) 1+sin(Θ/2)/parenrightBigg dφ/radicallow 1+sin(Θ/2)=  1/radicallow 1+sin(θ1/2)/radicallow 1+sin(θ2/2)K/parenleftBig/radicalBig 2sin( θ1/2) 1+sin(θ1/2)/parenrightBig K/parenleftBig/radicalBig 2sin( θ2/2) 1+sin(θ2/2)/parenrightBig ,θ1+θ2≤π 1/radicallow 1+cos(θ1/2)/radicallow 1+cos(θ2/2)K/parenleftBig/radicalBig 2cos( θ1/2) 1+cos(θ1/2)/parenrightBig K/parenleftBig/radicalBig 2cos( θ2/2) 1+cos(θ2/2)/parenrightBig ,θ1+θ2≥π(13) where cosΘ=cosθ1cosθ2+sinθ1sinθ2cosφ. In particular, we have /bracketleftbigg K/parenleftbigg sinθ 2/parenrightbigg/bracketrightbigg2 =/integraldisplay π/2 0K(sinθcosφ)dφ, ∀θ∈/bracketleftBig 0,π 2/bracketrightBig ,(12†) K/parenleftbigg sinθ 2/parenrightbigg K/parenleftbigg cosθ 2/parenrightbigg =/integraldisplay π/2 0K/parenleftbigg/radicalBig 1−sin2θcos2φ/parenrightbigg dφ, ∀θ∈/bracketleftBig 0,π 2/bracketrightBig ,(12‡) 1 1+/radicallow u/bracketleftBigg K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg2 =/integraldisplay π/2 0K/parenleftBigg/radicalBigg 4/radicallow u(1−u) cosφ 1+2/radicallow u(1−u) cosφ/parenrightBigg dφ/radicalbig 1+2/radicallow u(1−u) cosφ, ∀u∈/bracketleftbigg 0,1 2/bracketrightbigg ,(13†) /radicallow 2 1+/radicallow uK/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg =/integraldisplay π/2 0K/parenleftBigg/radicalBigg 2/radicallow 1−4u(1−u)cos2φ 1+/radicallow 1−4u(1−u)cos 2φ/parenrightBigg dφ/radicalbig 1+/radicallow 1−4u(1−u)cos2φ,∀u∈(0,1).(13‡) 4 Proof When ν=nis a non-negative integer, one can verify Eq. 11 by resorting to Eq. 10: 1 2π/integraldisplay 2π 0Pn(cosθ1cosθ2+sinθ1sinθ2cosφ)dφ =1 2π/integraldisplay 2π 0/bracketleftBigg 2 π/integraldisplay π 0θH(cosβ−cosθ1cosθ2−sinθ1sinθ2cosφ)cos[( n+1 2)β]dβ /radicalbig cosβ−cosθ1cosθ2−sinθ1sinθ2cosφ/bracketrightBigg dφ =1 2π/integraldisplay 2π 0/bracketleftBigg 2 π/integraldisplay π 04π∞/summationdisplay ℓ=0ℓ/summationdisplay m=−ℓYℓm(θ1,0)Yℓm(θ2,φ)cos[( ℓ+1 2)β]cos[( n+1 2)β] 2ℓ+1dβ/bracketrightBigg dφ=Pn(cosθ1)Pn(cosθ2), where spherical harmonic modes other than Yn0will have vanishing contribution upon integration in the an gular variables βandφ. Here, due to the parity relation Pn(−x)=(−1)nPn(x) for non-negative integers n, one always has Pn(cosθ1)Pn(cosθ2)=Pn(−cosθ1)Pn(−cosθ2), irrespective of the sum θ1+θ2. Fixing two positive parameters sandt, one may consider a neighborhood of the origin in the complex z-plane such that Recos( sz)>0,Recos( tz)>0,Re[cos( sz)cos( tz)+sin(sz)sin( tz)cosφ]>0. Then, given an arbitrary ν∈C, we can apply the Laplace integral representation for the Legendre functions (Eq. 4) to evaluate the following scaling limit: fs,t;ν(z) :=Pν/z(cos( sz))Pν/z(cos( tz))−/integraldisplay 2π 0Pν/z(cos( sz)cos( tz)+sin(sz)sin( tz)cosφ)dφ 2π =/integraldisplay 2π 0eν zlog(cos( sz)+isin(sz)cosφ1)dφ1 2π/integraldisplay 2π 0eν zlog(cos( tz)+isin(tz)cosφ2)dφ2 2π −/integraldisplay 2π 0/braceleftbigg/integraldisplay 2π 0eν zlog(cos( sz)cos( tz)+sin(sz)sin( tz)cosφ+i/radicallow 1−[cos( sz)cos( tz)+sin(sz)sin( tz)cosφ]2cosφ3)dφ3 2π/bracerightbiggdφ 2π →/integraldisplay 2π 0eiνscosφ1dφ1 2π/integraldisplay 2π 0eiνtcosφ2dφ2 2π−/integraldisplay 2π 0/braceleftbigg/integraldisplay 2π 0eiν/radicallow s2+t2−2stcosφcosφ3dφ3 2π/bracerightbiggdφ 2π=0,asz→0, where the last equality turns out to be a special case of the So nine-Gegenbauer formula for Bessel functions [Ref. 15, §12.1]: J0(νs)J0(νt)=1 2π/integraldisplay 2π 0J0(ν/radicalbig s2+t2−2stcosφ)dφ. Thus, with ﬁxed parameters s,tandν, the function fs,t;ν(z) is analytic near the origin in the complex z-plane, and fs,t;ν(z)=0 on the set {0}∪{ν/n|n∈Z∩[0,+∞)}, which contains an accumulation point {0}in the domain of analyticity forfs,t;ν(z). Therefore, the function fs,t;ν(z) must vanish identically in a certain neighborhood of the or igin, the spatial extent of which depends on the nature of the positive paramet erssandt. Now suppose that s+t<π, then cos scost+sinssintcosφ≥cos(s+t)> −1. Accordingly, for a complex number z sitting in a certain open neighborhood of the unit interval [ 0,1], the expression cos( sz)cos( tz)+sin(sz)sin( tz)cosφwill miss the value −1, which is the logarithmic branch point of the Legendre func tions Pµof non-integer degree µ∉Z. Hence, when s>0,t>0 and s+t<π, we have a univalent complex-analytic function fs,t;ν(z) in an open neighborhood of the unit interval [0 ,1], where resides a non-isolated set of zeros. By the princip le of analytic continuation, we may conclude that fs,t;ν(1)=0 for s>0,t>0 and s+t<π, which extends, by continuity, to the s+t=πscenario as well as those situations where st=0. This proves Eq. 11 for the cases of θ1+θ2≤π. The rest of Eq. 11 follows from the invariance of the integral 1 2π/integraldisplay 2π 0Pν(cosθ1cosθ2+sinθ1sinθ2cosφ)dφ under the equatorial reﬂections ( θ1,θ2)/mapstochar→(π−θ1,π−θ2). Setting ν=−1/2 in Eq. 11, we obtain Eq. 12; setting ν=−1/4 or ν=−3/4 in Eq. 11, one veriﬁes Eq. 13. When θ1=θ2, Eq. 12 (resp. 13) becomes Eq. 12†(resp. 13†). When θ1=π−θ2=θ, one may specialize Eq. 12 into Eq. 12‡. A similar procedure on Eq. 13 would bring us the evaluation /integraldisplay π/2 0K/parenleftBigg/radicalBigg 2/radicallow 1−4u(1−u)cos2φ 1+/radicallow 1−4u(1−u)cos 2φ/parenrightBigg dφ/radicalbig 1+/radicallow 1−4u(1−u)cos2φ =1/radicalbig 1+/radicallow u/radicalbig 1+/radicallow 1−uK/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg K/parenleftBigg/radicalBigg 2/radicallow 1−u 1+/radicallow 1−u/parenrightBigg ,∀u∈/parenleftbigg 0,1 2/bracketrightbigg , 5 while the substitution u=(1−t)2/(1+t)2and Landen’s transformation [Ref. 13, item 8.126] allow us t o see that 1/radicalbig 1+/radicallow 1−uK/parenleftBigg/radicalBigg 2/radicallow 1−u 1+/radicallow 1−u/parenrightBigg =/radicallow 1+t 1+/radicallow tK/parenleftbigg24/radicallow t 1+/radicallow t/parenrightbigg =/radicallow 1+tK(/radicallow t)=/radicalBigg 2 1+/radicallow uK/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg . This veriﬁes Eq. 13‡foru∈(0,1/2]. Furthermore, the right-hand side of Eq. 13‡is intact under the transformation u/mapstochar→1−u, whereas the symmetric extension of its left-hand side hing es on the identity 1 1+/radicallow uK/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg =1 1+/radicallow 1−uK/parenleftBigg/radicalBigg 1−/radicallow 1−u 1+/radicallow 1−u/parenrightBigg K/parenleftBigg/radicalBigg 2/radicallow 1−u 1+/radicallow 1−u/parenrightBigg ,0<u<1, (14) which is an equivalent formulation for the products of two La nden’s transformations: 1+t 2K(/radicallow 1−t)K(/radicallow t)=1+t (1+/radicallow t)2K/parenleftbigg1−/radicallow t 1+/radicallow t/parenrightbigg K/parenleftbigg24/radicallow t 1+/radicallow t/parenrightbigg ,0<t<1, (14′) with the correspondence of variables being u=(1−t)2/(1+t)2. /squaresolid Remark (1) Using Eq. 10, we may readily adapt our proof of the product formula in Eq. 11 to associated Legendre functions: 1 2πΓ(ν+m+1) Γ(ν−m+1)/integraldisplay 2π 0Pν(cosθ1cosθ2+sinθ1sinθ2cosφ)cosmφdφ=/braceleftBigg Pm ν(cosθ1)Pm ν(cosθ2), θ1+θ2≤π Pm ν(−cosθ1)Pm ν(−cosθ2),θ1+θ2≥π(11m) where θ1,θ2∈[0,π),ν∈C,m∈Z. Here, the corresponding scaling limit is the Sonine-Gegen bauer formula Jm(νs)Jm(νt)=1 2π/integraldisplay 2π 0J0(ν/radicalbig s2+t2−2stcosφ)cosmφdφ. (Doubtlessly, the cases of P1 −1/2(cosθ)=2 πsinθ/bracketleftbigg E/parenleftbigg sinθ 2/parenrightbigg −K/parenleftbigg sinθ 2/parenrightbigg cos2θ 2/bracketrightbigg , P1 −1/4(cosθ)=P1 −3/4(cosθ)=/radicallow 1+sin(θ/2) πsinθ/bracketleftBigg E/parenleftBigg/radicalBigg 2sin( θ/2) 1+sin(θ/2)/parenrightBigg −K/parenleftBigg/radicalBigg 2sin( θ/2) 1+sin(θ/2)/parenrightBigg/parenleftbigg 1−sinθ 2/parenrightbigg/bracketrightBigg will then lead to generalizations of Eqs. 12 and 13 into integ ral formulae involving complete elliptic integrals of the second kind.) Effectively, we can combine Eq. 11mwith Eq. 11 to develop a Fourier expansion in φfor the function Pν(cosθ1cosθ2+sinθ1sinθ2cosφ) (necessarily under the constraints 0 ≤θ1≤π,0≤θ2≤π,0≤θ1+θ2≤π), which turns up as a uniformly convergent series formerly known to E. W. Ho bson [Ref. 8, §226]: Pν(cosθ1cosθ2+sinθ1sinθ2cosφ)=Pν(cosθ1)Pν(cosθ2)+2∞/summationdisplay m=1Γ(ν−m+1) Γ(ν+m+1)Pm ν(cosθ1)Pm ν(cosθ2)cosmφ. Hence we refer to Eq. 11 as the Hobson coupling formula. Hobso n’s original approach draws on the Fourier series of (z+/radicallow z2−1cosφ)νfor complex-valued ν, which is not a method based on scaling limits. (2) The representation of Bessel functions Jmas the scaling limit of (associate) Legendre functions Pm νis a special case of Hansen’s formula [Ref. 15, §5.7]. Heine extended the limit procedure so that one may connect Bessel functions Ymand the (associate) Legendre functions Qm νin a similar fashion [Ref. 15, §5.71]. The Hansen-Heine scal ing limit procedure ( m=0) will be used again during the proof of Eq. 19 (ν,ν), namely, /integraldisplay 1 −1[Pν(x)]2Pν(−x)dx=limz→ν1+2cos( πz) 3πΓ/parenleftbigz+1 2/parenrightbig Γ/parenleftbig3z+2 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig3Γ/parenleftbig3z+3 2/parenrightbig,ν∈C in Proposition 2.1. (3) The integral formula in Eq. 12‡is usually attributed to Glasser [16], and has been mentione d recently in the work of Bailey et al. [2]. We will recover Eq. 12‡later in Proposition 3.3 (see §3.2), using a method independ ent of the Hobson coupling formula for Legendre functions. 6 (4) The Hobson coupling formula is sensitive to the geometri c constraint 0 ≤θ1+θ2≤π. Thus, it is not possible to directly extrapolate the formulae in this lemma to an integr al representation for [ Pν(x)]2,−1<x<0 where νis arbitrary. Practically, this obstacle can be overcome in various ways. Forν=−1/2,−1/3,−1/4,−1/6, the angular restriction in the Hobson coupling formula can be circumvented by using Ramanu jan’s integral representation for [ K(k)]2,0≤k<1 (see Proposition 3.3). For generic ν, Hobson’s approach yields an integral representation for Pν(x)Pν(−x),−1<x<1 and [Pν(x)]2,0≤x<1, and the ﬁnite Hilbert transform in Proposition 4.2 (see §4 .2) will enable us to express [ Pν(x)]2− [Pν(−x)]2,0≤x<1 in terms of Pν(ξ)Pν(−ξ),−1<ξ<1. /square 2 Generalized Clebsch-Gordan Integrals As the C2(−1,1) solutions to the Legendre differential equation of arbit rary degree ν∈C/integerdivideZ<0: d dx/bracketleftbigg (1−x2)df(x) dx/bracketrightbigg +ν(ν+1)f(x)=0 (15) are exhausted by linear combinations of Pν(x) and Qν(x), a special case in Appell’s theory of third order ordinary differential equations [17] then tells us that the C3(−1,1) solutions to d dx/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)df(x) dx/bracketrightbigg +4ν(ν+1)(1−x2)f(x)/bracerightbigg +4ν(ν+1)xf(x)=0,ν∈C/integerdivideZ<0 (16) belong to a three-dimensional space spanned by [ Pν(x)]2,Pν(x)Qν(x) and [ Qν(x)]2(or by an equivalent basis set {[Pν(x)]2,Pν(x)Pν(−x),[Pν(−x)]2}when ν∉Z). The “interpolation” from Legendre polynomials Pℓ(x),ℓ∈Z≥0to non-integer degree Legendre functions Pν(x),ν∉Z and the “scaling limit” for large degrees will enable us to ex tend some Clebsch-Gordan integral formulae to generic Legendre functions, as elaborated in the proposition below . Proposition 2.1 (Generalized Clebsch-Gordan Integrals) Deﬁne Tµ,ν:=/integraldisplay 1 −1Pµ(x)Pν(x)Pν(−x)dx,µ,ν∈C, then we have the symmetry Tµ,ν=T−µ−1,ν=Tµ,−ν−1=T−µ−1,−ν−1, (17) the recursion relation (µ+1)2[(µ+1)2−(2ν+1)2]Tµ+1,ν−µ2[µ2−(2ν+1)2]Tµ−1,ν=4(2µ+1)sin( µπ)sin(νπ) π2, (18) the representation in terms of generalized hypergeometric series Tµ,ν=2 π2sin(µπ)sin(νπ) µ(µ+1)/bracketleftBigg 1 ν4F3/parenleftBigg 1,1−µ 2,µ+2 2,−ν 2−µ 2,µ+3 2,1−ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/parenrightBigg −1 ν+14F3/parenleftBigg 1,1−µ 2,µ+2 2,ν+1 2−µ 2,µ+3 2,ν+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/parenrightBigg/bracketrightBigg ,µ,ν∉Z (19) Tµ,ν=2 πsin(µπ)cos(νπ) 2ν+1/bracketleftBigg 1 µ5F4/parenleftBigg1 2,1 2,−µ 2,−ν,1+ν 1,2−µ 2,1−2ν 2,2ν+3 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/parenrightBigg −1 µ+15F4/parenleftBigg1 2,1 2,1+µ 2,−ν,1+ν 1,3+µ 2,1−2ν 2,3+2ν 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/parenrightBigg/bracketrightBigg ,µ,ν∉Z,ν/negationslash=−1 2(19∗) along with some generalized Clebsch-Gordan integral formu lae: Tµ,n=(−1)nπΓ/parenleftBig2n+1−µ 2/parenrightBig Γ/parenleftBig2n+2+µ 2/parenrightBig /bracketleftBig Γ/parenleftBig1−µ 2/parenrightBig/bracketrightBig2/bracketleftBig Γ/parenleftBigµ+2 2/parenrightBig/bracketrightBig2 Γ/parenleftBig2n+2−µ 2/parenrightBig Γ/parenleftBig2n+3+µ 2/parenrightBig,µ∉Z,n∈Z≥0 (19(µ,n)) T2m,ν=πcos(νπ)Γ/parenleftbig2ν+1−2m 2/parenrightbig Γ(ν+1+m) /bracketleftbig Γ/parenleftbig1−2m 2/parenrightbig/bracketrightbig2[Γ(m+1)]2Γ(ν+1−m)Γ/parenleftbig2ν+3+2m 2/parenrightbig,m∈Z≥0,ν+1 2∉Z (19(2m,ν)) T2m,n+1 2=  −(−1)n π[Γ(m+1 2)]2Γ(m−n−1 2)Γ(m+n+3 2) [Γ(m+1)]2Γ(m−n)Γ(m+n+2),m∈Z≥0,n∈Z,m−n∉Z≤0,m+n+2∉Z≤0 0, m∈Z≥0,(m−n∈Z≤0orm+n+2∈Z≤0)(19(2m,n+1 2)) 7 Tν,ν=limz→ν1+2cos( πz) 3πΓ/parenleftbigz+1 2/parenrightbig Γ/parenleftbig3z+2 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig3Γ/parenleftbig3z+3 2/parenrightbig,ν∈C (19(ν,ν)) T2ν−1,ν=limz→νsin(πz)sin(2 πz) π2z2,ν∈C (19(2ν−1,ν)) T2ν+2,ν= −limz→νsin(πz)sin(2 πz) π2(z+1)2,ν∈C. (19(2ν+2,ν)) Proof The symmetry of Legendre functions with respect to degree Pν(x)=P−ν−1(x),∀ν∈Cexplains the related prop- erty of Tµ,ν(Eq. 17). Judging from the familiar recursion relations for Legendre functions (2µ+1)(1−x2)dPµ(x) dx=µ(µ+1)[Pµ−1(x)−Pµ+1(x)]; (2 µ+1)xPµ(x)=(µ+1)Pµ+1(x)+µPµ−1(x) and the differential equations given in Eqs. 15 and 16, we may carry out the following computations: ν(ν+1)[(µ+1)Tµ+1,ν+µTµ−1,ν] =ν(ν+1)/integraldisplay 1 −1[(µ+1)Pµ+1(x)+µPµ−1(x)]Pν(x)Pν(−x)dx=(2µ+1)ν(ν+1)/integraldisplay 1 −1xPµ(x)Pν(x)Pν(−x)dx = −2µ+1 4/integraldisplay 1 −1Pµ(x)d dx/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d(Pν(x)Pν(−x)) dx/bracketrightbigg +4ν(ν+1)(1−x2)Pν(x)Pν(−x)/bracerightbigg dx =2µ+1 4/integraldisplay 1 −1/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d(Pν(x)Pν(−x)) dx/bracketrightbigg +4ν(ν+1)(1−x2)Pν(x)Pν(−x)/bracerightbiggdPµ(x) dxdx =2µ+1 4/integraldisplay 1 −1/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d(Pν(x)Pν(−x)) dx/bracketrightbigg/bracerightbiggdPµ(x) dxdx+µ(µ+1)ν(ν+1)(Tµ−1,ν−Tµ+1,ν) =2µ+1 4/bracketleftbigg µ(µ+1)/integraldisplay 1 −1(1−x2)Pµ(x)d(Pν(x)Pν(−x)) dxdx−4sin( µπ)sin(νπ) π2/bracketrightbigg +µ(µ+1)ν(ν+1)(Tµ−1,ν−Tµ+1,ν), where we have used in the last line the properties Pν(1)=1,∀ν∈Cand lim x→−1+0+(1−x2)dPν(x) dx=2sin( νπ) π,∀ν∈C. We may then proceed with the simpliﬁcation (2µ+1)/integraldisplay 1 −1(1−x2)Pµ(x)d(Pν(x)Pν(−x)) dxdx = −(2µ+1)/integraldisplay 1 −1(1−x2)Pν(x)Pν(−x)dPµ(x) dxdx+2(2µ+1)/integraldisplay 1 −1xPµ(x)Pν(x)Pν(−x)dx =µ(µ+1)(Tµ+1,ν−Tµ−1,ν)+2(µ+1)Tµ+1,ν+2µTµ−1,ν, so as to conﬁrm the recursion formula of Tµ,ν(Eq. 18). For non-negative integers mand nmeeting the requirement m≤2n, we might recall from the standard Clebsch- Gordan theory that Tm,n:=/integraldisplay 1 −1Pm(x)Pn(x)Pn(−x)dx=2(−1)n/parenleftbiggm n n 0 0 0/parenrightbigg2 =(−1)nπΓ/parenleftbig2n+1−m 2/parenrightbig Γ/parenleftbig2n+2+m 2/parenrightbig /bracketleftbig Γ/parenleftbig1−m 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigm+2 2/parenrightbig/bracketrightbig2Γ/parenleftbig2n+2−m 2/parenrightbig Γ/parenleftbig2n+3+m 2/parenrightbig,(19(m,n)) where/parenleftbigj1j2j3m1m2m3/parenrightbig is the Wigner 3- jsymbol. The right-hand side expression in Eq. 19 (m,n)is regarded as zero if m is a positive odd integer, as anticipated from the relations Pm(x)= −Pm(−x) andΓ(1−m 2)= ∞ in such circumstances. We have Tm,n=/integraltext 1 −1Pm(x)Pn(x)Pn(−x)dx=0 for non-negative integers mandnsatisfying m>2n, as indicated by the respective poles of the gamma factors in the denominator on t he right-hand side of Eq. 19 (m,n). Actually, even without the prior knowledge of the standard C lebsch-Gordan theory, one can start from the initial condition T0,n=/integraltext 1 −1Pn(x)Pn(−x)dx=2(−1)n/(2n+1),n∈Z≥0, and build Eq. 19 (m,n)inductively from the recursion rela- tion given by Eq. 18. 8 The interpolation formula (known as Dougall’s expansion in [Ref. 18, p. 167]) Pν(x)=2/integraldisplay π 0θH(cosβ−x)cos(2ν+1)β 2/radicalbig 2(cos β−x)dβ π=2/integraldisplay π 0∞/summationdisplay ℓ=0Pℓ(x)cos(2ℓ+1)β 2cos(2ν+1)β 2dβ π =∞/summationdisplay ℓ=0Pℓ(x)/bracketleftbiggsin(ℓ−ν)π (ℓ−ν)π+sin(ℓ+ν+1)π (ℓ+ν+1)π/bracketrightbigg allows us to sum Eq. 19 (m,n)as Tµ,n:=/integraldisplay 1 −1Pµ(x)Pn(x)Pn(−x)dx=∞/summationdisplay ℓ=0Tℓ,n/bracketleftbiggsin(ℓ−µ)π (ℓ−µ)π+sin(ℓ+µ+1)π (ℓ+µ+1)π/bracketrightbigg forµ∈C/integerdivideZ,n∈Z≥0. To identify the last term in the equation above with the expr ession in Eq. 19 (µ,n), we note that the sum over ℓin fact truncates after ﬁnite terms, which reveals the expre ssion 1 sin(µπ)  (−1)nπΓ/parenleftBig2n+1−µ 2/parenrightBig Γ/parenleftBig2n+2+µ 2/parenrightBig /bracketleftBig Γ/parenleftBig1−µ 2/parenrightBig/bracketrightBig2/bracketleftBig Γ/parenleftBigµ+2 2/parenrightBig/bracketrightBig2 Γ/parenleftBig2n+2−µ 2/parenrightBig Γ/parenleftBig2n+3+µ 2/parenrightBig−∞/summationdisplay ℓ=0Tℓ,n/bracketleftbiggsin(ℓ−µ)π (ℓ−µ)π+sin(ℓ+µ+1)π (ℓ+µ+1)π/bracketrightbigg   as a rational function of µ, for whatever integer n. As µapproaches any integer m, it is clear that such a rational function tends to zero with O(µ−m) convergence rate, so it must be identically vanishing. By Eq. 11 along with the Mehler-Dirichlet theory, we have the following computation when νis not an integer: Pν(x)Pν(−x)=/integraldisplay 2π 0Pν((1−x2)cosφ−x2)dφ 2π=2/integraldisplay 2π 0/bracketleftBigg/integraldisplay π 0θH(cosβ−(1−x2)cosφ+x2)cos(2ν+1)β 2/radicalbig 2(cos β−(1−x2)cosφ+x2)dβ π/bracketrightBigg dφ 2π =2/integraldisplay 2π 0/bracketleftBigg/integraldisplay π 0∞/summationdisplay ℓ=0Pℓ((1−x2)cosφ−x2)cos(2ℓ+1)β 2cos(2ν+1)β 2dβ π/bracketrightBigg dφ 2π =∞/summationdisplay ℓ=0Pℓ(x)Pℓ(−x)/bracketleftbiggsin(ℓ−ν)π (ℓ−ν)π+sin(ℓ+ν+1)π (ℓ+ν+1)π/bracketrightbigg , where we have integrated over βandφseparately in the last step. Thus, we can perform a decomposi tion Tµ,ν=∞/summationdisplay ℓ=0Tµ,ℓ/bracketleftbiggsin(ℓ−ν)π (ℓ−ν)π+sin(ℓ+ν+1)π (ℓ+ν+1)π/bracketrightbigg =sin(νπ)(τµ,ν+τµ,−ν−1) /bracketleftBig Γ/parenleftBig1−µ 2/parenrightBig/bracketrightBig2/bracketleftBig Γ/parenleftBigµ+2 2/parenrightBig/bracketrightBig2, where τµ,λ:=∞/summationdisplay ℓ=0Γ/parenleftBig2ℓ+1−µ 2/parenrightBig Γ/parenleftBig2ℓ+2+µ 2/parenrightBig Γ/parenleftBig2ℓ+2−µ 2/parenrightBig Γ/parenleftBig2ℓ+3+µ 2/parenrightBig1 λ−ℓ=2sin( µπ) λπ2/bracketleftbigg Γ/parenleftbigg1−µ 2/parenrightbigg/bracketrightbigg2/bracketleftbigg Γ/parenleftbiggµ+2 2/parenrightbigg/bracketrightbigg2 4F3/parenleftBigg 1,1−µ 2,µ+2 2,−λ 2−µ 2,µ+3 2,1−λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/parenrightBigg follows directly from the deﬁnition of 4F3. This veriﬁes Eq. 19. One can prove Eq. 19 (2m,ν)by directly computing T0,ν=sin(νπ) π∞/summationdisplay n=0(−1)nT0,n/parenleftbigg1 ν−n−1 ν+n+1/parenrightbigg =sin(νπ) π∞/summationdisplay n=02 2n+1/parenleftbigg1 ν−n−1 ν+n+1/parenrightbigg =2cos( νπ) 2ν+1(19(0,ν)) and invoking the recursion relation in Eq. 18. Once Eq. 19 (2m,ν)is given, and T2m+1,ν=0,m∈Z≥0is obvious from symmetry considerations, we can interpolate Tµ,νas follows: Tµ,ν=∞/summationdisplay ℓ=0Tℓ,ν/bracketleftbiggsin(ℓ−µ)π (ℓ−µ)π+sin(ℓ+µ+1)π (ℓ+µ+1)π/bracketrightbigg =sin(µπ) π∞/summationdisplay m=0T2m,ν/parenleftbigg1 µ−2m−1 2m+µ+1/parenrightbigg =sin(µπ)cos(νπ)∞/summationdisplay m=0Γ/parenleftbig2ν+1−2m 2/parenrightbig Γ(ν+1+m) /bracketleftbig Γ/parenleftbig1−2m 2/parenrightbig/bracketrightbig2[Γ(m+1)]2Γ(ν+1−m)Γ/parenleftbig2ν+3+2m 2/parenrightbig/parenleftbigg1 µ−2m−1 2m+µ+1/parenrightbigg , which in turn, directly accounts for the hypergeometric sum mation in Eq. 19∗concerning 5F4. Exploiting again Euler’s reﬂection formula Γ(z)Γ(1−z)=π/sin(πz), one can rewrite the expression for T2m,νas (taking appropriate limits when the resulting fractions assume indeterminate forms) T2m,ν=−sin(νπ) π[Γ(m+1 2)]2Γ(m−ν)Γ(m+ν+1) [Γ(m+1)]2Γ(m−ν+1 2)Γ(m+ν+3 2),m∈Z≥0, 9 which specializes to Eq. 19(2m,n+1 2). To prove Eq. 19 (ν,ν), we show that the function T(z) :=1 sin2(πz)/braceleftBigg Tz,z−1+2cos( πz) 3πΓ/parenleftbigz+1 2/parenrightbig Γ/parenleftbig3z+2 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig3Γ/parenleftbig3z+3 2/parenrightbig/bracerightBigg ≡T(−z−1) (20) is analytic over the whole complex z-plane, with asymptotic behavior T(z)=O(N−9/4),z∈CN,N→+∞ . Here, Nis a positive integer and the square contour CNhas vertices (1 −4N)/4−iN, (1+4N)/4−iN, (1+4N)/4+iNand (1−4N)/4−iN. First, to verify that T(z) is an entire function, we only need to check that the express ion inside the braces of Eq. 20 has vanishing derivative whenever zis an integer, so that the numerator of Eq. 20 encounters (at l east) a second-order zero at every integer z=n∈Z. At the positive odd integers z=2n+1,n∈Z≥0, one may directly compute ∂Tz,z ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2n+1=2/integraldisplay 1 −1∂Pz(x) ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2n+1P2n+1(x)P2n+1(−x)dx+/integraldisplay 1 −1[P2n+1(−x)]2∂Pz(x) ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2n+1dx=∂Tz,2n+1 ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2n+1=0 from Eq. 19 (µ,n), which coincides with the behavior 1+2cos( πz) 3πΓ/parenleftbigz+1 2/parenrightbig Γ/parenleftbig3z+2 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig3Γ/parenleftbig3z+3 2/parenrightbig=O((z−(2n+1))2) attributed to the factor [ Γ((1−z)/2)]2=O((z−(2n+1))−2). Meanwhile, at the non-negative even integers z=2n,n∈Z≥0, ∂Tz,z ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2n=2/integraldisplay 1 −1∂Pz(x) ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2nP2n(x)P2n(−x)dx+/integraldisplay 1 −1[P2n(−x)]2∂Pz(x) ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2ndx=3∂Tz,2n ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2n =3T2n,2n∂ ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=2nlogΓ/parenleftbig4n+1−z 2/parenrightbig Γ/parenleftbig4n+2+z 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig2Γ/parenleftbig4n+2−z 2/parenrightbig Γ/parenleftbig4n+3+z 2/parenrightbig is also a result of Eq. 19 (µ,n). As we have 3∂ ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=νlogΓ/parenleftbig2ν+1−z 2/parenrightbig Γ/parenleftbig2ν+2+z 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig2Γ/parenleftbig2ν+2−z 2/parenrightbig Γ/parenleftbig2ν+3+z 2/parenrightbig−∂ ∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=νlogΓ/parenleftbigz+1 2/parenrightbig Γ/parenleftbig3z+2 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig3Γ/parenleftbig3z+3 2/parenrightbig=−2πtanνπ 2, which vanishes when ν=2nis a non-negative even number, it is clear that the numerator of Eq. 20 has O((z−2n)2) behavior for n∈Z≥0. Then, we investigate the asymptotic behavior of T(z) as z→ ∞ . For any point zsitting on the right half of the contour CN(where Nis a large positive integer) satisfying Re z>−1 2, we have a uniform asymptotic formula Tz,z=/bracketleftbigg 1+O/parenleftbigg1 N/parenrightbigg/bracketrightbigg/integraldisplay π/2 0/braceleftbigg J0/parenleftbigg/parenleftbigg z+1 2/parenrightbigg θ/parenrightbigg [1+cos(πz)]+Y0/parenleftbigg/parenleftbigg z+1 2/parenrightbigg θ/parenrightbigg sin(πz)/bracerightbigg J0/parenleftbigg/parenleftbigg z+1 2/parenrightbigg θ/parenrightbigg × ×/bracketleftbigg J0/parenleftbigg/parenleftbigg z+1 2/parenrightbigg θ/parenrightbigg cos(πz)+Y0/parenleftbigg/parenleftbigg z+1 2/parenrightbigg θ/parenrightbigg sin(πz)/bracketrightbigg/radicalBigg θ3 sinθdθ. (21) Here, we have used the following transformations for −π<argν<π: Tν,ν=/integraldisplay π 0[Pν(cosθ)]2Pν(−cosθ)sinθdθ=/integraldisplay π/2 0[Pν(cosθ)+Pν(−cosθ)]Pν(cosθ)Pν(−cosθ)sinθdθ =/integraldisplay π/2 0/braceleftbigg Pν(cosθ)[1+cos(νπ)]−2 πQν(cosθ)sin(νπ)/bracerightbigg Pν(cosθ)/bracketleftbigg Pν(cosθ)cos(νπ)−2 πQν(cosθ)sin(νπ)/bracketrightbigg sinθdθ, along with the asymptotic formulae (see [Ref. 19, Chap. 12, E qs. 12.18 and 12.25] or [Ref. 20, Eqs. 43 and 46]): Pν(cosθ)=/radicalBigg θ sinθJ0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg +O/parenleftbigg1 2ν+1/parenrightbigg ,Qν(cosθ)=−π 2/radicalBigg θ sinθY0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg +O/parenleftbigg1 2ν+1/parenrightbigg 10 where the error bounds are uniform for the angular variables θ∈(0,π/2], so long as |ν|→∞ ,−π<argν<π[19, 20]. For the case with degree ν=(1+4N)/4 where Nis a large positive integer, we split the integral on the righ t-hand side of Eq. 21 into two parts: /integraldisplay π/2 0(···)dθ=/integraldisplay 1//radicallow 2ν+1 0(···)dθ+/integraldisplay π/2 1//radicallow 2ν+1(···)dθ. For the ﬁrst portion, we may deduce /integraldisplay 1//radicallow 2ν+1 0/braceleftbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg [1+cos(νπ)]+Y0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg sin(νπ)/bracerightbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg × ×/bracketleftbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg cos(νπ)+Y0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg sin(νπ)/bracketrightbigg/radicalBigg θ3 sinθdθ=8[1+cos(νπ)][1+2cos( νπ)] 3/radicallow 3π(1+2ν)2/bracketleftbigg 1+O/parenleftbigg1 4/radicallow 2ν+1/parenrightbigg/bracketrightbigg from three integral formulae valid for a>0 [Ref. 21, items 2.12.42.25, 2.13.22.11, 2.13.25.1]:1 /integraldisplay ∞ 0[J0(ax)]3xdx=2/radicallow 3πa2,/integraldisplay ∞ 0[J0(ax)]2Y0(ax)xdx=0,/integraldisplay ∞ 0J0(ax)[Y0(ax)]2xdx=2 3/radicallow 3πa2, as well as the familiar asymptotic behavior of Bessel functi ons for large arguments x≫1: J0(x)∼/radicalBigg 2 πxcos/parenleftBig x−π 4/parenrightBig ,Y0(x)∼/radicalBigg 2 πxsin/parenleftBig x−π 4/parenrightBig , which allows us to verify /integraldisplay ∞ 1//radicallow 2ν+1/braceleftbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg x/parenrightbigg [1+cos(νπ)]+Y0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg x/parenrightbigg sin(νπ)/bracerightbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg x/parenrightbigg × ×/bracketleftbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg x/parenrightbigg cos(νπ)+Y0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg x/parenrightbigg sin(νπ)/bracketrightbigg xdx=O/parenleftbigg1 (2ν+1)9/4/parenrightbigg via integration by parts.2Meanwhile, using the asymptotic forms of Bessel functions a nd integration by parts, one can also verify that3 /integraldisplay π/2 1//radicallow 2ν+1/braceleftbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg [1+cos(νπ)]+Y0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg sin(νπ)/bracerightbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg × ×/bracketleftbigg J0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg cos(νπ)+Y0/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ/parenrightbigg sin(νπ)/bracketrightbigg/radicalBigg θ3 sinθdθ ∼/bracketleftbigg4 π(2ν+1)/bracketrightbigg3/2/integraldisplay π−(1//radicallow 2ν+1) 1//radicallow 2ν+1cos2/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ−π 4/parenrightbigg cos/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg (π−θ)−π 4/parenrightbiggdθ/radicallow sinθ=O/parenleftbigg1 (2ν+1)9/4/parenrightbigg . 1The ﬁrst integral here belongs to the generalized Weber-Sch afheitlin type [Ref. 15, §13.46]. The proof of the second int egral can be found in [22]. The third integral can be derived from the ﬁrst one, i n conjunction with Nicholson’s formula [Ref. 15, §13.73] [ J0(x)]2+[Y0(x)]2= 8 π2/integraltext ∞ 0K0(2xsinh t)dtand a special case of the modiﬁed Weber-Schafheitlin integr al [Ref. 15, §13.45]/integraltext ∞ 0J0(x)K0(2xsinh t)xdx=(1+4sinh2t)−1. 2To wit, we need to estimate /integraldisplay ∞ 1//radicallow 2ν+1sin(a(2ν+1)x+b)/radicallowxdx=sinb a/radicallow 2ν+1/integraldisplay ∞ /radicallow 2ν+11 /radicallowy∂ ∂ysin(ay)dy−cosb a/radicallow 2ν+1/integraldisplay ∞ /radicallow 2ν+11 /radicallowy∂ ∂ycos(ay)dy =cos(a/radicallow 2ν+1+b) a(2ν+1)3/4+sinb 2a/radicallow 2ν+1/integraldisplay ∞ /radicallow 2ν+1sin(ay) y3/2dy−cosb 2a/radicallow 2ν+1/integraldisplay ∞ /radicallow 2ν+1cos(ay) y3/2dy=O/parenleftbigg1 (2ν+1)3/4/parenrightbigg , where the coefﬁcients a,b∈Rarise from the product-to-sum formulae of trigonometric fu nctions applied to asymptotic expansions for the products o f Bessel functions. 3One might wish to compare this with the direct application of a hypergeometric asymptotic formula for Legendre function s (see [Ref. 8, p. 295] or [Ref. 18, p. 162]): Pν(cosθ)=Γ(ν+1) Γ(ν+3 2)/radicalBigg 2 πsinθ/bracketleftbigg cos/parenleftbigg/parenleftbigg ν+1 2/parenrightbigg θ−π 4/parenrightbigg +O/parenleftbigg1 ν/parenrightbigg/bracketrightbigg ,where θ∈[ε,π−ε],ε>0. 11 Thus, the asymptotic expansion Tν,ν=8[1+cos(νπ)][1+2cos( νπ)] 3/radicallow 3π(1+2ν)2/bracketleftbigg 1+O/parenleftbigg1 4/radicallow 2ν+1/parenrightbigg/bracketrightbigg is established for ν=(1+4N)/4>0. For an arbitrary point zsatisfying Re z>−1/2, we may modify the aforementioned procedure by deforming the integral path joining the points 0 and π/2 into the sum of two line segments: /integraldisplay π/2 0(···)dθ=/integraldisplay /radicallow|2z+1|/(2z+1) 0(···)dθ+/integraldisplay π/2 /radicallow|2z+1|/(2z+1)(···)dθ, followed by the integrations/integraltext ∞ 0(···)dxand/integraltext ∞/radicallow|2z+1|/(2z+1)(···)dxalong contours that run to inﬁnity in such a manner that (2 z+1)x→+∞ along the positive real axis. This results in Tz,z=8[1+cos(πz)][1+2cos( πz)] 3/radicallow 3π(1+2z)2/bracketleftbigg 1+O/parenleftbigg1 4/radicallow N/parenrightbigg/bracketrightbigg ,z∈CN,Rez>−1 2, (22a) where the error bound is uniform along the contour CN, on which both |tan(πz)|and|cot(πz)|are uniformly bounded. After reﬂection z/mapstochar→−z−1, we obtain Tz,z=8[1−cos(πz)][1−2cos( πz)] 3/radicallow 3π(1+2z)2/bracketleftbigg 1+O/parenleftbigg1 4/radicallow N/parenrightbigg/bracketrightbigg ,z∈CN,Rez<−1 2. (22b) We may combine Eqs. 22a and 22b into a single formula Tz,z=1+2cos( πz) 3πΓ/parenleftbigz+1 2/parenrightbig Γ/parenleftbig3z+2 2/parenrightbig /bracketleftbig Γ/parenleftbig1−z 2/parenrightbig/bracketrightbig2/bracketleftbig Γ/parenleftbigz+2 2/parenrightbig/bracketrightbig3Γ/parenleftbig3z+3 2/parenrightbig/bracketleftbigg 1+O/parenleftbigg1 4/radicallow N/parenrightbigg/bracketrightbigg ,z∈CN, which also implies the bound estimate T(z)=O(N−9/4) uniformly applicable to all the points zon the rectangular contour CN, thanks to the inequalities sup z∈CN|cot(πz)|≤max(1 ,cothπ)=cothπ,sup z∈CN1 |sin(πz)|≤max/parenleftbigg/radicallow 2,1 sinhπ/parenrightbigg =/radicallow 2, forN∈Z>0. By Cauchy’s integral formula, we have the identity T(ν)=lim N→∞/contintegraldisplay CNT(z) z−νdz 2πi=0,∀ν∈C, which proves Eq. 19 (ν,ν). Both Eqs. 19 (2ν−1,ν)and 19 (2ν+2,ν)follow directly from the recursion relation (Eq. 18). /squaresolid Remark As in our proof of Eq. 19 (ν,ν), we can use Bessel functions to establish the following asym ptotic behavior for |z|→∞ ,Rez>−1/2: Tρz,z∼8 πcos(ρπz+πz)+cos(πz) ρ/radicalbig 4−ρ2(1+2z)2+16 π2sin(ρπz)sin(πz) ρ/radicalbig 4−ρ2(1+2z)2 π−arcsin/radicalBigg 1−ρ2 4 , 0<ρ<2; Tρz,z∼8 πsin(ρπz−πz)−sin(πz) ρ/radicalbig ρ2−4(1+2z)2−16 π2sin(ρπz)sin(πz) ρ/radicalbig ρ2−4(1+2z)2sinh−1/radicalBigg ρ2 4−1, ρ>2. 12 along with an asymptotic reﬂection formula Tρ(−z−1),−z−1∼eiπ(1−ρ)arg z/|argz|Tρz,z,0<|argz|<πforρ>0,|z|→∞ . Put differently, for large |z|and−π<argz<π, we have an “asymptotic trigonometric modulation” of the st andard Clebsch- Gordan integral formulae: Tρz,z∼πΓ/parenleftBig1+(2−ρ)z 2/parenrightBig Γ/parenleftBig(ρ+2)z+2 2/parenrightBig /bracketleftBig Γ/parenleftBig1−ρz 2/parenrightBig/bracketrightBig2/bracketleftBig Γ/parenleftBig2+ρz 2/parenrightBig/bracketrightBig2 Γ/parenleftBig2+(2−ρ)z 2/parenrightBig Γ/parenleftBig(ρ+2)z+3 2/parenrightBig× × cos(ρπz+πz)+cos(πz) 1+cos(ρπz)+2 πsin(ρπz)sin(πz) 1+cos(ρπz) π−arcsin/radicalBigg 1−ρ2 4  , 0<ρ<2; Tρz,z∼πΓ/parenleftBig1+(2−ρ)z 2/parenrightBig Γ/parenleftBig(ρ+2)z+2 2/parenrightBig /bracketleftBig Γ/parenleftBig1−ρz 2/parenrightBig/bracketrightBig2/bracketleftBig Γ/parenleftBig2+ρz 2/parenrightBig/bracketrightBig2 Γ/parenleftBig2+(2−ρ)z 2/parenrightBig Γ/parenleftBig(ρ+2)z+3 2/parenrightBig× × sin(ρπz−πz)−sin(πz) 1+cos(ρπz)−2 πsin(ρπz)sin(πz) 1+cos(ρπz)sinh−1/radicalBigg ρ2 4−1 cot(ρ−2)πz 2, ρ>2. Forρ/negationslash=1, the right-hand side of the penultimate (resp. last) asymp totic formula diverges at z=−2/(2+ρ) (resp. z= −1/ρ), so the asymptotic analysis does not result in exact forms o fTρz,z(ρ/negationslash=1) for ﬁnitely sized |z|. /square Corollary 2.2 (Some Multiple Elliptic Integrals in Clebsch -Gordan Forms) We have the integral formulae π2 4T0,−1/2=2/integraldisplay 1 0K(/radicallow t)K(/radicallow 1−t)dt=π3 4, (23) π2 4T0,1/2=2/integraldisplay 1 0[2E(/radicallow t)−K(/radicallow t)][2E(/radicallow 1−t)−K(/radicallow 1−t)]dt=0, (24) π3 8T−1/2,−1/2=2/integraldisplay 1 0[K(/radicallow 1−t)]2K(/radicallow t)dt=[Γ(1 4)]8 192π2, (25) π3 8T−1/3,−1/3=27 2/integraldisplay 1 0(1−p2)p(2+p)/radicalbig 3+6p(1+p+p2)/bracketleftBigg K/parenleftBigg/radicalBigg p3(2+p) 1+2p/parenrightBigg/bracketrightBigg2 K/parenleftBigg/radicalBigg 1−p3(2+p) 1+2p/parenrightBigg dp =27 6/integraldisplay 1 0(1−p2)p(2+p)/radicalbig 1+2p(1+p+p2)/bracketleftBigg K/parenleftBigg/radicalBigg 1−p3(2+p) 1+2p/parenrightBigg/bracketrightBigg2 K/parenleftBigg/radicalBigg p3(2+p) 1+2p/parenrightBigg dp=3/radicallow 3[Γ(1 3)]9 256π2, (26) π3 8T−1/4,−1/4=/integraldisplay 1 04 (1+/radicallow u)3/2/bracketleftBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg2 K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg du=4/radicallow 2/integraldisplay 1 0(1−t)[K(/radicallow t)]2K(/radicallow 1−t) (1+t)3/2dt =/integraldisplay 1 0/parenleftbigg2 1+/radicallow u/parenrightbigg3/2/bracketleftBigg K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg2 K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg du=4/integraldisplay 1 0(1−t)[K(/radicallow 1−t)]2K(/radicallow t) (1+t)3/2dt=[Γ(1 8)Γ(3 8)]2 24,(27) π3 8T−1/6,−1/6=27 2/radicallow 2/integraldisplay 1 −1/bracketleftBigg K/parenleftBigg/radicalbigg 1−x 2/parenrightBigg/bracketrightBigg2 K/parenleftBigg/radicalbigg 1+x 2/parenrightBigg 1−x2 (3+x2)7/4dx=27 4/integraldisplay 1 0t(1−t)[K(/radicallow 1−t)]2K(/radicallow t) (1−t+t2)7/4dt=[Γ(1 4)]4 8/radicalbig 2/radicallow 3,(28) π3 8T1/2,−3/4=2/radicallow 2/integraldisplay 1 02E(/radicallow u)−K(/radicallow u) 1+/radicallow uK/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg du=/radicallow 2π, (29) π3 8T3/2,−1/4=2/radicallow 2 3/integraldisplay 1 08(1−2u)E(/radicallow u)−(5−8u)K(/radicallow u) 1+/radicallow uK/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg du=−/radicallow 2π 9. (30) Proof Special cases of Eqs. 19(2m,n+1 2), 19 (ν,ν)and 19 (2ν+2,ν)lead to Eqs. 23, 24, 25, 26, 27, 28, 29 and 30. /squaresolid 3 Spherical Rotations and Multiple Elliptic Integrals In this section, we shall focus on the transformations of mul tiple elliptic integrals using variable substitutions motivated by rotations on a unit sphere. In effect, our atten tion will be momentarily restricted to two special Legendre functions P−1/2andP−1/4. 13 The transformation methods in this section not only provide evaluations of certain Cauchy principal values: P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)d ξ π(x−ξ)/radicalbig 1+ξ,−1<x<1; P/integraldisplay 1 −1K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg 2dξ π(x−ξ),−1<x<1 etc. which will be used in §4, but also pave way for further applica tions in our subsequent works on elliptic integrals. 3.1 Beltrami Rotations The technique of spherical rotation, which traces back to Be ltrami’s work on the Abel integral equations [Ref. 23, p. 328], has found applications in the geometrically-motiv ated evaluation of certain deﬁnite integrals related to Bes sel functions (see [Ref. 15, §3.33, §12.12 and §12.14]). The lem ma below is a simple realization of the Beltrami rotation in the case of the complete elliptic integrals K(k) and E(k). Lemma 3.1 (Beltrami Transformations) (a)We have the following integral identities K(k)=2 π/integraldisplay 1 0K(/radicallow 1−κ2)dκ 1−k2κ2, 0<k<1, (31) K(ξ)=2 π/integraldisplay 1 0/radicalbig 1−ξ2K(/radicallow 1−κ2)dκ 1−ξ2(1−κ2), 0<ξ<1, (31′) K(r)=2 π/integraldisplay 1 0(1+r)K(/radicallow 1−κ2)dκ (1+r)2−4rκ2, 0<r<1, (31L) K(η)=2 π/integraldisplay 1 0(1−η)K(/radicallow 1−κ2)dκ (1−η)2+4ηκ2, 0<η<1. (31′ L) (b)For0<k<1, we have K(k)−E(k) k2=2 π/integraldisplay 1 0E(/radicallow 1−κ2)dκ 1−k2κ2, (32) E(k)=4(1−k2) π/integraldisplay 1 0E(/radicallow 1−κ2)dκ (1−k2κ2)2, (33) E(k)/radicallow 1−k2=4(1−k2) π/integraldisplay 1 0E(/radicallow 1−κ2)dκ [1−k2(1−κ2)]2. (33′) Proof (a) We start by writing the complete elliptic integral of the ﬁrst kind as an integral on the unit sphere S2: K(k)=1 2/integraldisplay π 0dθ/radicalbig 1−k2sin2θ=1 4π/integraldisplay π 0sinθdθ/integraldisplay 2π 0dφ1 sinθ(1−ksinθcosφ)=/integraldisplay S21/radicallow 1−Z2(1−kX)dσ 4π. As we rotate about the Y-axis by a right angle, effectively swapping the rôles of the X- and Z-axes, we may rewrite the integral above as K(k)=/integraldisplay S21/radicallow 1−X2(1−kZ)dσ 4π=/integraldisplay S21/radicallow 1−X2(1−k2Z2)dσ 4π=1 4π/integraldisplay π 0sinθdθ/integraldisplay 2π 0dφ1/radicalbig 1−sin2θcos2φ(1−k2cos2θ), where we have combined the values of the integrand at points ±Zand restored the spherical coordinates. Integrating over the azimuthal angle φ, we obtain K(k)=1 π/integraldisplay π 0K(sinθ)sinθdθ 1−k2cos2θ, which is equivalent to Eq. 31. Now that we have proved the relation K(k)=2 π/integraldisplay 1 0K(/radicallow 1−κ2) 1−k2κ2dκ,0<k<1, which analytically continues to k∈C/integerdivide[1,+∞), we may derive Eqs. 31′, 31 Land 31′ L, respectively, from the imagi- nary modulus transformation K(ξ)=K(iξ//radicalbig 1−ξ2)//radicalbig 1−ξ2, Landen’s transformation K(r)=K(2/radicallowr/(1+r))/(1+r), and a combination thereof K(η)=K(2i/radicallowη/(1−η))/(1−η). 14 (b) We adapt our derivation of part (a) to accommodate for the complete elliptic integrals of the second kind E(k),0< k<1: K(k)−E(k) k2=1 2/integraldisplay π 0sin2θdθ/radicalbig 1−k2sin2θ=1 4π/integraldisplay π 0sinθdθ/integraldisplay 2π 0dφsinθ 1−ksinθcosφ=/integraldisplay S2/radicallow 1−Z2 1−kXdσ 4π =/integraldisplay S2/radicallow 1−X2 1−kZdσ 4π=/integraldisplay S2/radicallow 1−X2 1−k2Z2dσ 4π=1 4π/integraldisplay π 0sinθdθ/integraldisplay 2π 0dφ/radicalbig 1−sin2θcos2φ 1−k2cos2θ=1 π/integraldisplay π 0E(sinθ)sinθdθ 1−k2cos2θ, which veriﬁes Eq. 32. As we multiply both sides of Eq. 32 by k2, and differentiate in k, we obtain the formula stated in Eq. 33: kE(k) 1−k2=d dk[K(k)−E(k)]=2 πd dk/integraldisplay 1 0E(/radicallow 1−κ2)k2dκ 1−k2κ2=4k π/integraldisplay 1 0E(/radicallow 1−κ2)dκ (1−k2κ2)2,0<k<1. Now, by analytic continuation, we have E(k)=4(1−k2) π/integraldisplay 1 0E(/radicallow 1−κ2)dκ (1−k2κ2)2,k∈C/integerdivide[1,+∞). (33∗) In particular, by the imaginary modulus transformation for complete elliptic integrals of the second kind [Ref. 24, item 160.02], we obtain /radicalbig 1+ξ2E/parenleftBigg ξ/radicalbig 1+ξ2/parenrightBigg =E(iξ)=4(1+ξ2) π/integraldisplay 1 0E(/radicallow 1−κ2)dκ (1+ξ2κ2)2,ξ>0, which is equivalent to Eq. 33′. /squaresolid Remark Admittedly, the spherical rotation is not the only road to an y or all of the identities in Eqs. 31, 31′, 31 Land 31′ L. For example, one may still verify Eq. 31 via the method of mom ents and hypergeometric summations [6]. The same can be said for several formulae that we will encounter in Pro position 3.3. However, it is our hope that the spherical rotations provide clearer geometric motivations than a pur ely combinatorial approach. The author thanks an anonymous referee for pointing out a rec ent contribution of V. Anghel [25], which generalizes the Beltrami rotation to hyperspheres embedded in Euclidea n spaces of arbitrary dimensions. It is highly probable that the hyperspherical analog of Beltrami rotation can lea d to geometric proofs of various integral identities relate d to elliptic integrals, which we hope to pursue in a separate p ublication. /square Beltrami transformations allow us to convert one multiple e lliptic integral into another. For instance, the following chain of identities /integraldisplay 1 0K(ξ)dξ=/integraldisplay 1 0K(/radicallow 1−κ2) 1+κdκ=/integraldisplay 1 0K(ξ)ξ 1+/radicalbig 1−ξ2dξ/radicalbig 1−ξ2=2 π/integraldisplay 1 0K/parenleftBig/radicalbig 1−κ2/parenrightBig/bracketleftbiggκarccos κ/radicallow 1−κ2−log(2κ)/bracketrightbigg dκ result from two applications of Eq. 31′, and the transformations /integraldisplay 1 0E(k)dk=2 π/integraldisplay 1 0E(/radicallow 1−κ2) κ3[−κ+(1+κ2)tanh−1κ]dκ=/integraldisplay 1 0(2+κ)E(/radicallow 1−κ2) (1+κ)2dκ can be justiﬁed by Eqs. 33 and 33′. More examples will be described elsewhere. In the next corollary, we mention a few consequences of Beltr ami transformations that are pertinent to later devel- opments in the current work. Corollary 3.2 (Some Applications of Beltrami Transformati ons) (a)For0<u<1, we have the following duality relations for multiple elliptic integrals: /integraldisplay /radicallow u 0K(k)dk/radicallow 1−k2/radicallow u−k2=/integraldisplay 1 0kK(k)dk/radicallow 1−k2/radicallow 1−k2u, (34) /integraldisplay 1 /radicallow ukK(k)dk/radicallow 1−k2/radicallow k2−u=/integraldisplay 1 0K(/radicallow 1−κ2)dκ/radicallow 1−κ2/radicallow 1−κ2u. (35) 15 (b)We have the following Cauchy principal values for −1<x<1: P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2) π(x−ξ)/radicalbig 1+ξdξ=K(/radicallow (1+x)/2)/radicallow 1+x, (36) P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)−2E(/radicalbig (1−ξ)/2) π(x−ξ)/radicalbig 1+ξdξ=2E(/radicallow (1+x)/2)−K(/radicallow (1+x)/2)/radicallow 1+x. (37) Proof (a) With Eq. 31′, we may interchange the order of integrations to compute /integraldisplay /radicallow u 0K(k)dk/radicallow 1−k2/radicallow u−k2=/integraldisplay /radicallow u 0/bracketleftBigg 2 π/integraldisplay 1 0/radicalbig 1−ξ2K(/radicallow 1−κ2)dκ 1−ξ2(1−κ2)/bracketrightBigg dξ/radicalbig 1−ξ2/radicalbig u−ξ2=/integraldisplay 1 0K(/radicallow 1−κ2)dκ/radicalbig 1−u(1−κ2), where the last expression is equivalent to the right-hand si de of Eq. 34, by the correspondence κ/mapstochar→/radicallow 1−k2. Similarly, we may prove Eq. 35 using Eq. 31: /integraldisplay 1 /radicallow ukK(k)dk/radicallow 1−k2/radicallow k2−u=/integraldisplay /radicallow u 0/bracketleftBigg 2 π/integraldisplay 1 0K(/radicallow 1−κ2)dκ 1−k2κ2/bracketrightBigg kdk/radicallow 1−k2/radicallow k2−u=/integraldisplay 1 0K(/radicallow 1−κ2)dκ/radicallow 1−κ2/radicallow 1−κ2u. (b) For z>1, one can readily compute (cf. [Ref. 26, p. 233]) K(/radicallowz)=/integraldisplay 1 0dt/radicallow 1−t2/radicallow 1−zt2=/integraldisplay 1//radicallowz 0dt/radicallow 1−t2/radicallow 1−zt2−i/integraldisplay 1 1//radicallowzdt/radicallow 1−t2/radicallow zt2−1 =1/radicallowz/integraldisplay 1 0ds/radicalBig 1−1 zs2/radicallow 1−s2−i/radicallowz/integraldisplay /radicallowz 1ds/radicalBig 1−1 zs2/radicallow s2−1=K(/radicallow 1/z)−iK(/radicallow (z−1)/z)/radicallowz. (38) This is the inverse modulus transformation for complete ell iptic integrals of the ﬁrst kind. Now, in the identity K(k)=2 π/integraldisplay 1 0K(/radicallow 1−κ2)dκ 1−k2κ2,∀k∈C/integerdivide[1,+∞), we approach the limit of k±i0+∈[1,+∞), read off the real part, and employ the inverse modulus tran sformation (Eq. 38), to obtain K(k)=k πP/integraldisplay 1 −1K(/radicallow 1−κ2)dκ k2−κ2,0<k<1, which can be reformulated into Eq. 36 after the variable subs titutions k=/radicallow (1+x)/2,κ=/radicalbig (1+ξ)/2. From Eq. 36, we may directly compute (cf. [Ref. 27, Eq. 11.215 ]) P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)/radicalbig 1+ξdξ π(x−ξ)=P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)[1+(ξ−x)+x]dξ π(x−ξ)/radicalbig 1+ξ =(1+x)P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)d ξ π(x−ξ)/radicalbig 1+ξ−1 π/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)d ξ/radicalbig 1+ξ =/radicallow 1+xK/parenleftBigg/radicalbigg 1+x 2/parenrightBigg −1 π/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)d ξ/radicalbig 1+ξ,−1<x<1. Here, the last integral can be simpliﬁed with the knowledge o f/integraltext 1 0K(/radicallow 1−κ2)dκ=π2/4 [Ref. 13, item 6.141.2], which brings us P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)/radicalbig 1+ξdξ π(x−ξ)=/radicallow 1+xK/parenleftBigg/radicalbigg 1+x 2/parenrightBigg −π/radicallow 2,−1<x<1. 16 Applying integration by parts to the Tricomi transform (cf. [Ref. 27, Eqs. 11.218 and 11.219]), one can use the equation above to deduce that P/integraldisplay 1 −11 π(x−ξ)d dξ/bracketleftBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg/radicalbig 1+ξ/bracketrightBigg dξ=d dξ/bracketleftBigg /radicallow 1+xK/parenleftBigg/radicalbigg 1+x 2/parenrightBigg/bracketrightBigg +1/radicallow 2(x−1),−1<x<1. In other words, we have P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)−E(/radicalbig (1−ξ)/2) π(x−ξ)(1−ξ)/radicalbig 1+ξdξ=1 1−x/bracketleftbiggE(/radicallow (1+x)/2)/radicallow 1+x−1/radicallow 2/bracketrightbigg ,−1<x<1. Carrying this further, we obtain P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)−E(/radicalbig (1−ξ)/2) π(x−ξ)/radicalbig 1+ξdξ=E(/radicallow (1+x)/2)/radicallow 1+x−1/radicallow 2+1 π/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)−E(/radicalbig (1−ξ)/2) (1−ξ)/radicalbig 1+ξdξ =E(/radicallow (1+x)/2)/radicallow 1+x−1/radicallow 2+1 π/integraldisplay 1 −1d dξ/bracketleftBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg/radicalbig 1+ξ/bracketrightBigg dξ=E(/radicallow (1+x)/2)/radicallow 1+x,−1<x<1. We may combine the relation above with Eq. 36 into a symmetric form P/integraldisplay 1 −1K(/radicalbig (1−ξ)/2)−2E(/radicalbig (1−ξ)/2) π(x−ξ)/radicalbig 1+ξdξ=2E(/radicallow (1+x)/2)−K(/radicallow (1+x)/2)/radicallow 1+x,−1<x<1, as stated in Eq. 37. /squaresolid 3.2 Ramanujan Rotations The next proposition is based on reverse engineering of some formulae in Ramanujan’s notebooks that were stated without proofs. Proposition 3.3 (Ramanujan Transformations) (a)Fors,t∈[0,1), we may represent K(/radicallow s)K(/radicallow t)and [K(/radicallow t)]2as integrals on the unit sphere S2: K(/radicallow s)K(/radicallow t)=1 8/integraldisplay S2dσ/radicalbig (1−sX2)(1−tY2)−Z2, (39) [K(/radicallow t)]2=/integraldisplay S22(2−t)K(/radicallow X2+Y2) (2−t)2−t2X2dσ 4π(40) =/integraldisplay S2(1+t)K(/radicallow X2+Y2) (1+t)2−4tX2dσ 4π, (40L) where X=sinθcosφ,Y=sinθsinφ,Z=cosθare the Cartesian coordinates on the unit sphere S2. Consequently, the following identities hold for 0<u<1: /integraldisplay /radicallow u 0K(k)dk/radicallow 1−k2/radicallow u−k2=/integraldisplay 1 0kK(k)dk/radicallow 1−k2/radicallow 1−k2u=1 1+/radicallow u/bracketleftBigg K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg2 , (41) /integraldisplay /radicallow 1−u 0K(k)dk/radicallow 1−k2/radicallow 1−u−k2=/integraldisplay 1 0kK(k)dk /radicallow 1−k2/radicalbig 1−k2(1−u)=2 1+/radicallow u/bracketleftBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg2 . (42) (b)We have the integral identity /integraldisplay S2K/parenleftBig/radicalbig X2+Y2/parenrightBig f(|X|)dσ 4π=2 π/integraldisplay 1 0f(k)K /radicalBigg 1+k 2 K /radicalBigg 1−k 2 dk (43) 17 so long as the function f(|X|),0≤|X|≤1assumes non-negative values and the surface integral is con vergent. This allows us to convert Eq. 40 into the following identities for 0<t<1: [K(/radicallow t)]2=2 π/integraldisplay 1 0K(/radicalbig µ)K(/radicalbig 1−µ)dµ 1−µt, (40∗) [K(/radicallow 1−t)]2=8 π/integraldisplay 1 0K(/radicalbig µ)K(/radicalbig 1−µ)dµ (1+/radicallow t)2−µ(1−/radicallow t)2, (40∗ L) and to deduce the following integral formulae:4 /integraldisplay 1 0K(/radicallow 1−κ2)/radicalbig (2t−1)2−(1−κ2)dκ=[K(/radicallow 1−t)−iK(/radicallow t)]2 2,0<t<1 2; (44) /integraldisplay 1 0K(/radicallow 1−κ2)/radicalbig (2t−1)2−(1−κ2)dκ=[K(/radicallow 1−t)+iK(/radicallow t)]2 2,t<0. (44′) Moreover, we have the identity below for 0<λ≤1/2: /integraldisplay 1 1−2λkK(k)dk /radicallow 1−k2/radicalbig k2−(1−2λ)2=K(/radicallow λ)K(/radicallow 1−λ)=/integraldisplay 1 (1−λ)/(1+λ)K(k)dk /radicallow 1−k2/radicalbig (1+λ)2k2−(1−λ)2, (45) or equivalently, /integraldisplay π/2 0K(sinθ)dθ/radicalbig 1−(1−2λ)2cos2θ=K(/radicallow λ)K(/radicallow 1−λ)=/integraldisplay π/2 0dθ/integraldisplay π/2 0dφ/radicalbig (1+λ)2−(1−λ)2sin2θ−4λsin2φ. (45′) Proof (a) Writing d σ=sinθdθdφfor the surface element on the unit sphere, we have K(/radicallows)K(/radicallow t)=1 8/integraldisplay π 0dθ/radicallow 1−scos2θ/integraldisplay 2π 0dφ/radicalbig 1−tcos2φ=1 8/integraldisplay S2dσ/radicallow 1−sZ2/radicallow 1−Z2−tX2 where the Cartesian coordinates X=sinθcosφ,Z=cosθwere used in the last step. As in Lemma 3.1, we interchange the rôles of the X- and Z-axes, to deduce K(/radicallow s)K(/radicallow t)=1 8/integraldisplay S2dσ/radicallow 1−sX2/radicallow 1−X2−tZ2=/integraldisplay π/2 0dθ/integraldisplay π/2 0dφsinθ/radicalbig 1−ssin2θcos2φ/radicalbig 1−sin2θcos2φ−tcos2θ =/integraldisplay π/2 0sinθdθ/radicallow 1−tcos2θ/integraldisplay π/2 0dφ/radicalBig 1−ssin2θcos2φ/radicalBig 1−sin2θ 1−tcos2θcos2φ. For any two numbers a,b∈(0,1), we can verify the identity /integraldisplay π/2 0dφ/radicalbig 1−acos2φ/radicalbig 1−bcos2φ=/integraldisplay π/2 0dψ/radicalbig 1−a−(b−a)cos2ψ(46) with a simple substitution φ=arctan(/radicallow 1−atanψ). Now, setting a=ssin2θ,b=sin2θ 1−tcos2θ, we are led to the formula K(/radicallow s)K(/radicallow t)=/integraldisplay π/2 0sinθdθ/integraldisplay π/2 0dφ/radicalbig (1−s)(1−sin2θcos2φ)+(s−t)cos2θ+stcos2θsin2θsin2φ =1 8/integraldisplay S2dσ/radicalbig (1−s)(1−X2)+(s−t)Z2+stY2Z2=1 8/integraldisplay S2dσ/radicalbig (1−sY2)(1−tZ2)−X2(39′) 4Here, in Eq. 44, we have chosen the univalent branches of the s quare roots such that Im/radicalbig (2t−1)2−(1−κ2)≥0, and K(/radicallow 1−t)≥0,K(/radicallow t)≥0; in Eq. 44′, it is understood that/radicalbig (2t−1)2−(1−κ2)≥0 on the left-hand side and K(/radicallow t)=/integraldisplay π/2 0dθ/radicalbig 1+|t|cos2θ≥0,ImK(/radicallow 1−t)=Im/integraldisplay π/2 0dθ/radicalbig 1−(1−t)cos2θ≤0,∀t<0. 18 after a literal transformation from the spherical coordina tes to Cartesian coordinates, and an algebraic simpliﬁcati on according to the spherical constraint X2+Y2+Z2=1. Clearly, the last term in Eq. 39′is equivalent to the right-hand side of Eq. 39, as the integral in question remains invariant under a cyclic shift of variables ( X,Y,Z)/mapstochar→(Z,X,Y). When s=t, we may swap the X- and Z-axes in the penultimate term of Eq. 39′, to derive [K(/radicallow t)]2=1 8/integraldisplay S2dσ/radicalbig (1−t)(1−Z2)+t2Y2X2=/integraldisplay π/2 0dθ/integraldisplay π/2 0dφ1/radicalbig 1−t+t2sin2θsin2φcos2φ =1 2/integraldisplay π/2 0dθ/integraldisplay π 0dφ1/radicalBig 1−t+t2 4sin2θsin2φ=1 8/integraldisplay S2dσ/radicalbigg (1−Z2)/parenleftBig 1−t+t2 4Y2/parenrightBig. Switching the Y- and Z-axes, we then arrive at [K(/radicallow t)]2=1 8/integraldisplay S2dσ/radicalbigg (1−Y2)/parenleftBig 1−t+t2 4Z2/parenrightBig=/integraldisplay π/2 0sinθdθ/integraldisplay π/2 0dφ1/radicalBig 1−sin2θsin2φ/radicalBig 1−t+t2 4cos2θ =/integraldisplay π/2 0K(sinθ)sinθdθ/radicalBig 1−t+t2 4cos2θ=/integraldisplay π 0K(sinθ)sinθdθ/radicalbig (2−t)2−t2sin2θ. Judging from the familiar integral formula /integraldisplay 2π 0dφ 2−t+tsinθcosφ=2π/radicalbig (2−t)2−t2sin2θ, it is evident that [K(/radicallow t)]2=1 4π/integraldisplay 2π 0dφ/integraldisplay π 02K(sinθ)sinθdθ 2−t+tsinθcosφ=1 4π/integraldisplay S22K(/radicallow X2+Y2)dσ 2−t+tX. Pairing up the values of the integrand at ±X, we obtain [K(/radicallow t)]2=1 4π/integraldisplay S22K(/radicallow X2+Y2)dσ 2−t+tX=2−t 4π/integraldisplay S22K(/radicallow X2+Y2)dσ (2−t)2−t2X2, as claimed in Eq. 40. Then, Landen’s transformation K(/radicallow t)=1 1+/radicallow tK/parenleftbigg24/radicallow t 1+/radicallow t/parenrightbigg ,0≤t<1 (47) brings us to Eq. 40 L. On account of Eqs. 40 and 40 L, we may put down the relations /bracketleftBigg K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg2 =(1+/radicallow u)/integraldisplay S2K(/radicallow X2+Y2) 1−uX2dσ 4π=(1+/radicallow u)/integraldisplay 1 0kK(k)dk/radicallow 1−k2/radicallow 1−k2u, /bracketleftBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg2 =1+/radicallow u 2/integraldisplay S2K(/radicallow X2+Y2) 1−(1−u)X2dσ 4π=1+/radicallow u 2/integraldisplay 1 0kK(k)dk /radicallow 1−k2/radicalbig 1−k2(1−u), which conﬁrm, respectively, the right halves of Eqs. 41 and 4 2. The left halves of Eqs. 41 and 42 follow from the duality relation in Eq. 34. One may wish to compare Eqs. 40 and 40 L(applicable to 0 <u<1) to the Hobson coupling formula in Eq. 13† (conﬁned to 0 ≤u≤1/2). (b) We start with a spherical rotation /integraldisplay S2K/parenleftBig/radicalbig X2+Y2/parenrightBig f(|X|)dσ 4π=/integraldisplay S2K/parenleftBig/radicalbig Z2+Y2/parenrightBig f(|Z|)dσ 4π =2 π/integraldisplay 1 0dZ/integraldisplay π/2 0dφK/parenleftbigg/radicalBig Z2+(1−Z2)sin2φ/parenrightbigg f(|Z|)=2 π/integraldisplay 1 0dk/integraldisplay π/2 0dφK/parenleftbigg/radicalBig k2cos2φ+sin2φ/parenrightbigg f(k) =2 π/integraldisplay 1 0f(k)/bracketleftBigg/integraldisplay /radicallow 1−k2 0K(/radicallow 1−κ2)/radicallow 1−k2−κ2dκ/bracketrightBigg dk=2 πRe/integraldisplay 1 0f(k)/bracketleftBigg/integraldisplay 1 0K(/radicallow 1−κ2)/radicallow 1−k2−κ2dκ/bracketrightBigg dk, 19 where we have made the substitution κ=/radicallow 1−k2cosφin the last line. Then, we may evaluate Re/bracketleftBigg/integraldisplay 1 0K(/radicallow 1−κ2)/radicallow 1−k2−κ2dκ/bracketrightBigg =−Im/bracketleftBigg/integraldisplay 1 0K(/radicallow 1−κ2)/radicallow k2−1+κ2dκ/bracketrightBigg by analytic continuation. From Eq. 40, we have /integraldisplay 1 0K(/radicallow 1−κ2)/radicalBig 1−z+z2 4κ2dκ=/bracketleftbigg/integraldisplay 1 0dt/radicallow 1−t2/radicallow 1−zt2/bracketrightbigg2 ,0≤z<1. After setting z=2/(1+k)≥1 in the inverse modulus transformation formula (Eq. 38), we obtain (1+k)/integraldisplay 1 0K(/radicallow 1−κ2)/radicallow k2−1+κ2dκ=1+k 2 K /radicalBigg 1+k 2 −iK /radicalBigg 1−k 2  2 ,0≤k<1. (48) Reading off the imaginary parts on both sides of the formula a bove, we arrive at the following equation for λ=(1−k)/2∈ (0,1/2]: /integraldisplay 1 1−2λkK(k)dk/radicalbig 1−k2/radicalbig k2−(1−2λ)2=K(/radicallow λ)K(/radicallow 1−λ), (49) as well as the identity claimed in Eq. 43. Applying Eq. 43 to Eq. 40, we obtain the result stated in Eq. 40∗: [K(/radicallow t)]2=/integraldisplay S22(2−t)K(/radicallow X2+Y2) (2−t)2−t2X2dσ 4π =2 π /integraldisplay 1 01 2−t+ktK /radicalBigg 1+k 2 K /radicalBigg 1−k 2 dk+/integraldisplay 1 01 2−t−k′tK /radicalBigg 1+k′ 2 K /radicalBigg 1−k′ 2 dk′  =2 π/bracketleftBigg/integraldisplay 1//radicallow 2 02K(/radicalbig µ)K(/radicalbig 1−µ) 2−t+(1−2µ)tdµ+/integraldisplay 1 1//radicallow 22K(/radicalbig µ′)K(/radicalbig 1−µ′) 2−t−(2µ′−1)tdµ′/bracketrightBigg =2 π/integraldisplay 1 0K(/radicalbig µ)K(/radicalbig 1−µ)dµ 1−µt. (An equivalent form of Eq. 40∗has appeared as Eq. 26 in [6], which was derived combinatoria lly using generalized hypergeometric series, instead of the geometric interpret ation given here.) One may derive Eq. 40∗ Lfrom Eq. 40∗and Landen’s transformation K(/radicallow 1−t)=2 1+/radicallow tK/parenleftbigg1−/radicallow t 1+/radicallow t/parenrightbigg . Writing t=(1−k)/2∈(0,1/2), we may recast Eq. 48 into Eq. 44. By analytically contin uing Eq. 44 to negative valued t<0 with the aid of the integral representation of the elliptic integral Kfor complex-valued modulus, we arrive at Eq. 44′. Here, the change in the sign from −iK(/radicallow t) in Eq. 44 to +iK(/radicallow t) in Eq. 44′is worthy of special attention. It is ready to appreciate that such a sign change ensures compatib ility with the natural choices of the univalent branches of the square roots occurring in all the places. The leftmost equality in Eq. 45 has been already proved in Eq. 49, and its counterpart in Eq. 45′follows from Eq. 35. To demonstrate the rightmost equality in Eq. 45, we perform t he following computations for 0 <λ≤1/2: /integraldisplay 1 (1−λ)/(1+λ)K(k)dk /radicallow 1−k2/radicalbig (1+λ)2k2−(1−λ)2=Re/integraldisplay π/2 0K(sinθ)dθ/radicalbig 4λ−(1+λ)2cos2θ=1 2/radicallow λRe/bracketleftBigg K/parenleftBigg i(1−/radicallow λ) 24/radicallow λ/parenrightBigg K/parenleftBigg (1+/radicallow λ) 24/radicallow λ/parenrightBigg/bracketrightBigg by using an analytic continuation of the leftmost term of Eq. 45′in the last step. According to the imaginary modulus transformation and Landen’s transformation, we have K/parenleftBigg i(1−/radicallow λ) 24/radicallow λ/parenrightBigg =24/radicallow λ 1+/radicallow λK/parenleftBigg 1−/radicallow λ 1+/radicallow λ/parenrightBigg =4/radicallow λK(/radicallow 1−λ); 20 whereas the inverse modulus transformation (Eq. 38) and Lan den’s transformation lead us to Re/bracketleftBigg K/parenleftBigg (1+/radicallow λ) 24/radicallow λ/parenrightBigg/bracketrightBigg =24/radicallow λ 1+/radicallow λK/parenleftBigg 24/radicallow λ 1+/radicallow λ/parenrightBigg =24/radicallow λK(/radicallow λ). Thus, the rightmost equality in Eq. 45 is veriﬁed. The connec tion to its counterpart in Eq. 45′can be proved as follows: /integraldisplay 1 /radicallow uK(k)dk/radicallow 1−k2/radicallow k2−u=/integraldisplay π/2 01/radicalbig 1−(1−u)sin2θK/parenleftBigg/radicalBigg u 1−(1−u)sin2θ/parenrightBigg dθ =/integraldisplay π/2 0dθ/integraldisplay π/2 0dφ/radicalbig 1−(1−u)sin2θ−usin2φ. Here, we have used the variable substitution k=/radicalbig u/[1−(1−u)sin2θ] to complete the ﬁrst line in the equation above, before spelling out K(/radicalbig u/[1−(1−u)sin2θ]) as an integral in φin the second line. The last equality in Eq. 45′implies the following evaluation for 0 <u<1: /integraldisplay π/2 0dθ/integraldisplay π/2 0dφ/radicalbig 1−usin2θ−(1−u)sin2φ=2 1+/radicallow uK/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg , (45′∗) so an interchange of the angular variables θandφbrings us back to the u↔1−usymmetry displayed in Eq. 14. One may also wish to compare the formula 2 1+/radicallow uK/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg =/integraldisplay 1 /radicallow uK(k)dk/radicallow 1−k2/radicallow k2−u,0<u<1 (50) to the corresponding result from Hobson coupling (Eq. 13‡). /squaresolid Remark (1) Up to a variable substitution, our intermediate result i n part (a) [K(/radicallow t)]2=/integraldisplay π/2 0K(sinθ)sinθdθ/radicalBig 1−t+t2 4cos2θ=/integraldisplay 1 0K(/radicallow 1−κ2)dκ/radicalBig 1−t+t2 4κ2,0≤t<1 (40′) is related to formula 13.0.18 in [28], and item 2.16.5.7 of [2 9]. However, it is highly probable that the ultimate source of this formula is Ramanujan. Entry 7(x) in Chapter 17 of Ramanujan’s second notebook esse ntially reads /integraldisplay π/2 0/integraldisplay π/2 0dθdφ/radicalbig (1−usin2θ)(1−usin2θsin2φ)=/parenleftBigg/integraldisplay π/2 0dψ/radicalbig 1−usin4ψ/parenrightBigg2 , which was veriﬁed by B. C. Berndt with some highly technical t ransformations of the generalized hypergeometric series 4F3[Ref. 11, pp. 110-111]. In fact, the left-hand side of Ramanu jan’s formula equals /integraldisplay π/2 0K(/radicallow usinθ)dθ/radicalbig 1−usin2θ=/integraldisplay π/2 0K(sinθ)sinθdθ/radicalbig 1−usin2θ after integration in φand reference to Eq. 34, while its right-hand side equals π2[P−1/4(1−2u)]2/4, according to the inte- gral identity in Eq. 6. In this manner, we see that Ramanujan’ s formula is actually equivalent to Eq. 40′, a consequence of spherical rotations. In [Ref. 11, pp. 111-112], Berndt followed up with a combinat orial proof of Entry 7(xi) in Ramanujan’s notebook: /integraldisplay π/2 0/integraldisplay π/2 0ksinθdθdφ/radicalbig (1−k2sin2θ)(1−k2sin2θsin2φ)=/integraldisplay π/2 0/integraldisplay arcsin k 0dθdφ/radicalbig 1−k2sin2θ−sin2θsin2φ =1 2   K /radicalBigg 1+k 2  2 − K /radicalBigg 1−k 2  2  , 21 which turns out to be exactly the real part of our Eq. 44, a geom etrically motivated result. Berndt has described his proofs of these two entries as “undo ubtedly not those found by Ramanujan”. Judging from the use of spherical coordinates in Ramanujan’s presentati on, we take leave to think that the technique of spherical rotations given in the proof of this proposition might be clo ser to a rediscovery of the pathway that Ramanujan has originally undertaken. (2) Combining our analysis in part (b) with Eq. 46, we have eff ectively shown that spherical geometry entails the following integral formula /integraldisplay π/2 0K/parenleftbigg/radicalBig k2cos2φ+sin2φ/parenrightbigg dφ=/integraldisplay π/2 0K(sinθ)dθ/radicallow 1−k2cos2θ=K /radicalBigg 1+k 2 K /radicalBigg 1−k 2 , which is a relation that has nearly 80 years of history. In 200 8, Bailey et al. used the method of Bessel moments to discover the following integral identity (Eq. 49 in [2]) /integraldisplay π/2 0K(sinθ)dθ/radicallow 1−cos2αcos2θ=K/parenleftBig sinα 2/parenrightBig K/parenleftBig cosα 2/parenrightBig , and remarked on its equivalence to a formula derived by Glass er in 1976: /integraldisplay π/2 0K/parenleftbigg/radicalBig 1−sin2αcos2φ/parenrightbigg dφ=K/parenleftBig sinα 2/parenrightBig K/parenleftBig cosα 2/parenrightBig . As pointed out by Zucker [3], the left-hand side in the formul a of Glasser, being a type of generalized Watson integral, can be expressed in terms of F4, an Appell hypergeometric function, which in turn, reduces to the product of two complete elliptic integrals of the ﬁrst kind, according to a result by Bailey in 1933 [30]. Needless to say, all these displayed formulae are equivalent to each other, and echo ba ck to Eq. 12‡, which was derived earlier from the Hobson coupling formula. /square Corollary 3.4 (Some Cauchy Principal Values) For−1<x<1, we have P/integraldisplay 1 −1K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg 2dξ π(x−ξ)=/bracketleftBigg K/parenleftBigg/radicalBigg 1+x 2/parenrightBigg/bracketrightBigg2 −/bracketleftBigg K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg/bracketrightBigg2 (51) and P/integraldisplay 1 −1/bracketleftBigg K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg +K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg E/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg −K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg E/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg/bracketrightBigg dξ π(x−ξ) = −/bracketleftBigg K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg/bracketrightBigg2 +K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg E/parenleftBigg/radicalBigg 1−x 2/parenrightBigg +K/parenleftBigg/radicalBigg 1+x 2/parenrightBigg E/parenleftBigg/radicalBigg 1+x 2/parenrightBigg . (52) Proof From Eq. 40∗and the inverse modulus transformation Re {[K(1//radicallow t)]2/t}=[K(/radicallow t)]2−[K(/radicallow 1−t)]2fort∈(0,1) (cf. Eq. 38), we may deduce [K(/radicallow t)]2−[K(/radicallow 1−t)]2=2 πP/integraldisplay 1 0K(/radicalbig µ)K(/radicalbig 1−µ)dµ t−µ,0<t<1, which is equivalent to Eq. 51. Now, we proceed as in the proof of Corollary 3.2(b), and compu te P/integraldisplay 1 −1K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg 2(1+ξ)dξ π(x−ξ)=(1+x)  /bracketleftBigg K/parenleftBigg/radicalBigg 1+x 2/parenrightBigg/bracketrightBigg2 −/bracketleftBigg K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg/bracketrightBigg2  −2 π/integraldisplay 1 −1K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg dξ =(1+x)  /bracketleftBigg K/parenleftBigg/radicalBigg 1+x 2/parenrightBigg/bracketrightBigg2 −/bracketleftBigg K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg/bracketrightBigg2  −π2 2, (53) 22 where we have quoted the result T0,−1/2=/integraltext 1 −1P−1/2(ξ)P−1/2(−ξ)dξ=πfrom a limit scenario of Eq. 19 (0,ν). Likewise, we may deduce from the equation above another Cauchy principal value: P/integraldisplay 1 −1K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg 2(1−ξ2)dξ π(x−ξ)=(1−x2)  /bracketleftBigg K/parenleftBigg/radicalBigg 1+x 2/parenrightBigg/bracketrightBigg2 −/bracketleftBigg K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg/bracketrightBigg2  +π2x 2. Like what we did in the proof of Corollary 3.2(b), we can use th e last equation to derive P/integraldisplay 1 −1d dξ/bracketleftBigg (1−ξ2)K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg/bracketrightBigg 2dξ π(x−ξ)=d dx (1−x2)  /bracketleftBigg K/parenleftBigg/radicalBigg 1+x 2/parenrightBigg/bracketrightBigg2 −/bracketleftBigg K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg/bracketrightBigg2  +π2x 2 .(54) after integration by parts. A little algebra then reveals th at Eqs. 53 and 54 together entail Eq. 52. /squaresolid 4 Finite Hilbert Transform and Tricomi Pairing 4.1 Parseval Identity for Tricomi Transforms The ﬁnite Hilbert transform on the interval ( −1,1), also known as the Tricomi transform, is deﬁned via a Cauch y principal value (see [Ref. 31, Chap. 4] or [Ref. 27, Chap. 11] ): (/hatwiderTf)(x) :=P/integraldisplay 1 −1f(ξ)dξ π(x−ξ),a.e.x∈(−1,1). This induces a continuous linear operator /hatwiderT:Lp(−1,1)−→Lp(−1,1) for 1<p<+∞ [Ref. 32, p. 188]. We can derive some new integral relations from a Parseval-ty pe identity for the Tricomi transform (see [Ref. 31, §4.3, Eq. 2] or [Ref. 27, Eq. 11.237]): /integraldisplay 1 −1f(x)(/hatwiderTg)(x)dx+/integraldisplay 1 −1g(x)(/hatwiderTf)(x)dx=0, (55) where f∈Lp(−1,1),p>1;g∈Lq(−1,1),q>1 and1 p+1 q<1. We call such a procedure “Tricomi pairing”, for which an example in the proposition below puts ﬁnishing touches on a p roof for the conjectural identity stated in the introductio n. Proposition 4.1 (An Application of Tricomi Pairing to Multi ple Elliptic Integrals) There are several integrals that evaluate to the same number [Γ(1 4)]8/(128 π2): /integraldisplay 1 0/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig3 dk=10 3/integraldisplay 1 0[K(k)]3dk=5/integraldisplay 1 0[K(k)]3kdk =3/integraldisplay 1 0[K(k)]2K/parenleftBig/radicalbig 1−k2/parenrightBig dk=2/integraldisplay 1 0K(k)/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig2 dk=6/integraldisplay 1 0[K(k)]2K/parenleftBig/radicalbig 1−k2/parenrightBig kdk. (56) Proof If we set f(x)=K(/radicallow (1−x)/2)//radicallow 1+x,−1<x<1 and g(x)=2K(/radicallow (1+x)/2)K(/radicallow (1−x)/2),−1<x<1 in Eq. 55, while recalling the Cauchy principal values given in Eqs. 36 and 51 , then we arrive at 2/integraldisplay 1 −1/bracketleftBigg K/parenleftBigg/radicalBigg 1+x 2/parenrightBigg/bracketrightBigg2 K/parenleftBigg/radicalBigg 1−x 2/parenrightBigg dx/radicallow 1+x=−/integraldisplay 1 −1K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg  /bracketleftBigg K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg/bracketrightBigg2 −/bracketleftBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg/bracketrightBigg2  dξ/radicalbig 1+ξ. This instantly rearranges into/integraltext 1 0[K(/radicallow 1−k2)]3dk=3/integraltext 1 0[K(k)]2K(/radicallow 1−k2)dk(an identity that Wan conjectured numer- ically in [6] without an analytic proof), which reveals the e quivalence between the leading items of the ﬁrst two lines in Eq. 56. Writing k=(1−ξ)/(1+ξ) and employing Landen’s transformation K(2/radicalbig ξ/(1+ξ))=(1+ξ)K(ξ), one has /integraldisplay 1 0/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig3 dk=/integraldisplay 1 0 K /radicalBigg 1−/parenleftbigg1−ξ 1+ξ/parenrightbigg2  3 d1−ξ 1+ξ=2/integraldisplay 1 0[K(ξ)]3(1+ξ)dξ; 23 writing k=(1−ξ)/(1+ξ) and employing Landen’s transformation 2 K((1−ξ)/(1+ξ))=(1+ξ)K(/radicalbig 1−ξ2), one has /integraldisplay 1 0[K(k)]3dk=/integraldisplay 1 0/bracketleftbigg K/parenleftbigg1−ξ 1+ξ/parenrightbigg/bracketrightbigg3 d1−ξ 1+ξ=1 4/integraldisplay 1 0/bracketleftbigg K/parenleftbigg/radicalBig 1−ξ2/parenrightbigg/bracketrightbigg3 (1+ξ)dξ=1 4/integraldisplay 1 0/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig3 dk+1 4/integraldisplay 1 0[K(η)]3ηdη, where the last step is a trivial substitution ξ/mapstochar→/radicalbig 1−η2. The two simultaneous equations displayed above make it pos - sible to eliminate any one among the three quantities/integraltext 1 0[K(/radicallow 1−k2)]3dk,/integraltext 1 0[K(k)]3dk,/integraltext 1 0[K(k)]3kdk, and determine the ratio between the two remaining numbers. Thus, we have ve riﬁed the chain of identities in the ﬁrst line of Eq. 56 (cf. [Ref. 6, Eq. 29]). The relations within the second line of Eq. 56 can be likewise established by succe ssive Landen’s transformations, as detailed in the ﬁrst paragraph of [Ref. 6, p. 139]. As we recall from Eq. 25 that 6/integraldisplay 1 0[K(k)]2K/parenleftBig/radicalbig 1−k2/parenrightBig kdk=[Γ(1 4)]8 128π2, the veriﬁcation is complete. /squaresolid Remark Sometimes, the output of Tricomi pairing can also be immedia tely recovered by more straightforward means. For example, upon setting f(ξ)=2K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg , g(ξ)=K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg +K/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg E/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg −K/parenleftBigg/radicalBigg 1+ξ 2/parenrightBigg E/parenleftBigg/radicalBigg 1−ξ 2/parenrightBigg in Eq. 55, one arrives at a vanishing identity: 0=/integraldisplay 1 0/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig2/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig K(k)+K/parenleftBig/radicalbig 1−k2/parenrightBig E(k)−3K(k)E/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig kdk. As pointed out by an anonymous referee, the equation above is anticipated from the fact that the integrand is precisely the derivative of k2(1−k2)K(k)[K(/radicallow 1−k2)]3, which makes it less surprising than Eq. 56. It might be still interesting to ask if Eq. 56 can be likewise reduced into ﬁnite steps of algeb raic manipulations on elliptic integrals and applications of the Newton-Leibniz formula (cf. §5). /square 4.2 Tricomi Transform of Pν(x)Pν(−x) The formula in Eq. 51 can be rewritten as P/integraldisplay 1 −12P−1/2(ξ)P−1/2(−ξ) π(x−ξ)dξ=−{[P−1/2(x)]2−[P−1/2(−x)]2},−1<x<1. This is not accidental. In the next proposition, we will gene ralize such a Tricomi transform relation to Legendre functions of arbitrary degree ν. Proposition 4.2 (Tricomi Transform of Pν(ξ)Pν(−ξ))For any ν∈C/integerdivideZ, we have P/integraldisplay 1 −12Pν(ξ)Pν(−ξ) π(x−ξ)dξ=[Pν(x)]2−[Pν(−x)]2 sin(νπ),−1<x<1. (57) Forn∈Z≥0, there is an identity P/integraldisplay 1 −1[Pn(ξ)]2 2(x−ξ)dξ=P/integraldisplay 1 −1[P−n−1(ξ)]2 2(x−ξ)dξ=Pn(x)Qn(x),−1<x<1. (58) Proof To verify Eq. 57, it would sufﬁce to demonstrate that 0=/integraldisplay 1 −1Pℓ(x)/braceleftbigg[Pν(x)]2−[Pν(−x)]2 sin(νπ)−P/integraldisplay 1 −12Pν(ξ)Pν(−ξ) π(x−ξ)dξ/bracerightbigg dx =1 sin(νπ)/integraldisplay 1 −1Pℓ(x){[Pν(x)]2−[Pν(−x)]2}dx+4 π/integraldisplay 1 −1Qℓ(x)Pν(x)Pν(−x)dx,∀ℓ∈Z≥0. (59) 24 Here, in the last line, we have used the Parseval identity of T ricomi pairing (Eq. 55), along with the Neumann integral representation for Legendre functions of the second kind (c f. [Ref. 27, Eq. 11.269] or [Ref. 33, Table 1.12A, Eq. 12A.26] ): Qℓ(x)=P/integraldisplay 1 −1Pℓ(ξ)dξ 2(x−ξ),∀x∈(−1,1),∀ℓ∈Z≥0. (60) For a non-negative even number ℓ, both addends in the last line of Eq. 59 vanish because the int egrands are odd functions. We may now settle Eq. 59 for odd numbers ℓby induction. For ℓ=1, we use Eq. 16 to compute 1 sin(νπ)/integraldisplay 1 −1P1(x){[Pν(x)]2−[Pν(−x)]2}dx=1 sin(νπ)/integraldisplay 1 −1x{[Pν(x)]2−[Pν(−x)]2}dx = −1 4ν(ν+1)sin( νπ)lim x→1−0+(1−x2)d dx/bracketleftbigg (1−x2)d{[Pν(x)]2−[Pν(−x)]2} dx/bracketrightbigg +1 4ν(ν+1)sin( νπ)lim x→−1+0+(1−x2)d dx/bracketleftbigg (1−x2)d{[Pν(x)]2−[Pν(−x)]2} dx/bracketrightbigg =4sin( νπ) ν(ν+1)π2. Meanwhile, we may check that 4 π/integraldisplay 1 −1Q1(x)Pν(x)Pν(−x)dx=4 π/integraldisplay 1 −1/parenleftbigg −1+x 2log1+x 1−x/parenrightbigg Pν(x)Pν(−x)dx = −4 π/integraldisplay 1 −1Pν(x)Pν(−x)dx−2 π/integraldisplay 1 −1log1+x 1−x 4ν(ν+1)d dx/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d(Pν(x)Pν(−x)) dx/bracketrightbigg +4ν(ν+1)(1−x2)Pν(x)Pν(−x)/bracerightbigg dx =4 π/integraldisplay 1 −11 4ν(ν+1)d dx/bracketleftbigg (1−x2)d(Pν(x)Pν(−x)) dx/bracketrightbigg dx=−4sin( νπ) ν(ν+1)π2. Thus, Eq. 59 holds for ℓ=1. Noting that the Legendre functions of the ﬁrst and second k inds satisfy similar recursion relations, namely, (2µ+1)(1−x2)dPµ(x) dx=µ(µ+1)[Pµ−1(x)−Pµ+1(x)]; (2 µ+1)xPµ(x)=(µ+1)Pµ+1(x)+µPµ−1(x); (2µ+1)(1−x2)dQµ(x) dx=µ(µ+1)[Qµ−1(x)−Qµ+1(x)]; (2 µ+1)xQµ(x)=(µ+1)Qµ+1(x)+µQµ−1(x), we can deduce ν(ν+1)(ℓ+1) sin(νπ)/integraldisplay 1 −1Pℓ+1(x){[Pν(x)]2−[Pν(−x)]2}dx+4ν(ν+1)(ℓ+1) π/integraldisplay 1 −1Qℓ+1(x)Pν(x)Pν(−x)dx +ν(ν+1)ℓ sin(νπ)/integraldisplay 1 −1Pℓ−1(x){[Pν(x)]2−[Pν(−x)]2}dx+4ν(ν+1)ℓ π/integraldisplay 1 −1Qℓ−1(x)Pν(x)Pν(−x)dx =ν(ν+1)(2ℓ+1) sin(νπ)/integraldisplay 1 −1xPℓ(x){[Pν(x)]2−[Pν(−x)]2}dx+4ν(ν+1)(2ℓ+1) π/integraldisplay 1 −1xQℓ(x)Pν(x)Pν(−x)dx, as well as 4ν(ν+1) 2ℓ+1/integraldisplay 1 −1[(ℓ+1)Pℓ+1(x)+ℓPℓ−1(x)]{[Pν(x)]2−[Pν(−x)]2}dx=4ν(ν+1)/integraldisplay 1 −1xPℓ(x){[Pν(x)]2−[Pν(−x)]2}dx = −/integraldisplay 1 −1Pℓ(x)d dx/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d{[Pν(x)]2−[Pν(−x)]2} dx/bracketrightbigg +4ν(ν+1)(1−x2)[Pν(x)]2−4ν(ν+1)(1−x2)[Pν(x)]2/bracerightbigg dx =/integraldisplay 1 −1/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d{[Pν(x)]2−[Pν(−x)]2} dx/bracketrightbigg +4ν(ν+1)(1−x2)[Pν(x)]2−4ν(ν+1)(1−x2)[Pν(x)]2/bracerightbiggdPℓ(x) dxdx +8[1+(−1)ℓ]sin2(νπ) π2 =ℓ(ℓ+1)/integraldisplay 1 −1(1−x2)Pℓ(x)d{[Pν(x)]2−[Pν(−x)]2} dxdx+4ℓ(ℓ+1)ν(ν+1) 2ℓ+1/integraldisplay 1 −1[Pℓ−1(x)−Pℓ+1(x)]{[Pν(x)]2−[Pν(−x)]2}dx +8[1+(−1)ℓ]sin2(νπ) π2, 25 which leads to a recursion relation (ℓ+1)2[(ℓ+1)2−(2ν+1)2]/integraldisplay 1 −1Pℓ+1(x){[Pν(x)]2−[Pν(−x)]2}dx −ℓ2[ℓ2−(2ν+1)2]/integraldisplay 1 −1Pℓ−1(x){[Pν(x)]2−[Pν(−x)]2}dx=−8[1+(−1)ℓ](2ℓ+1)sin2(νπ) π2, and 4ν(ν+1) 2ℓ+1/integraldisplay 1 −1[(ℓ+1)Qℓ+1(x)+ℓQℓ−1(x)]Pν(x)Pν(−x)dx=4ν(ν+1)/integraldisplay 1 −1xQℓ(x)Pν(x)Pν(−x)dx = −/integraldisplay 1 −1Qℓ(x)d dx/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d(Pν(x)Pν(−x)) dx/bracketrightbigg +4ν(ν+1)(1−x2)Pν(x)Pν(−x)/bracerightbigg dx =/integraldisplay 1 −1/braceleftbigg (1−x2)d dx/bracketleftbigg (1−x2)d(Pν(x)Pν(−x)) dx/bracketrightbigg +4ν(ν+1)(1−x2)Pν(x)Pν(−x)/bracerightbiggdQℓ(x) dxdx =ℓ(ℓ+1)/integraldisplay 1 −1(1−x2)Qℓ(x)d(Pν(x)Pν(−x)) dxdx+4ℓ(ℓ+1)ν(ν+1) 2ℓ+1/integraldisplay 1 −1[Qℓ−1(x)−Qℓ+1(x)]Pν(x)Pν(−x)dx −2[1+(−1)ℓ]sin(νπ) π, which brings us another recursion relation (ℓ+1)2[(ℓ+1)2−(2ν+1)2]/integraldisplay 1 −1Qℓ+1(x)Pν(x)Pν(−x)dx −ℓ2[ℓ2−(2ν+1)2]/integraldisplay 1 −1Qℓ−1(x)Pν(x)Pν(−x)dx=2[1+(−1)ℓ](2ℓ+1)sin( νπ) π. Here, we have used the identities lim x→−1+0+(1−x2)d dx/bracketleftbigg 2(1−x2)Pν(x)dPν(x) dx/bracketrightbigg =8sin2(νπ) π2;Pℓ(1)=(−1)ℓPℓ(−1)=1,∀ℓ∈Z≥0 to take care of boundary contributions to the integral conce rning Pℓ(x), and resorted to the limit behavior lim x→1−0+(1−x2)2dQµ(x) dxd(Pν(x)Pν(−x)) dx= −2sin( νπ) π lim x→−1+0+(1−x2)2dQµ(x) dxd(Pν(x)Pν(−x)) dx=2cos( µπ)sin(νπ) π for the integration by parts involving Qℓ(x). Therefore, the last line of Eq. 59 satisﬁes a homogeneous r ecursion (ℓ+1)2[(ℓ+1)2−(2ν+1)2]/braceleftbigg1 sin(νπ)/integraldisplay 1 −1Pℓ+1(x){[Pν(x)]2−[Pν(−x)]2}dx+4 π/integraldisplay 1 −1Qℓ+1(x)Pν(x)Pν(−x)dx/bracerightbigg =ℓ2[ℓ2−(2ν+1)2]/braceleftbigg1 sin(νπ)/integraldisplay 1 −1Pℓ−1(x){[Pν(x)]2−[Pν(−x)]2}dx+4 π/integraldisplay 1 −1Qℓ−1(x)Pν(x)Pν(−x)dx/bracerightbigg ,ℓ∈Z≥0 and the truthfulness of Eq. 59 for ℓ=1 entails any scenario with a larger odd number ℓ. This completes the veriﬁcation of Eq. 57 for ν∈C/integerdivideZ. For a ﬁxed x∈(−1,1), both sides of Eq. 57 represent continuous functions of ν, so we may handle Eq. 58 by investi- gating the ν→nlimit where n∈Z≥0. Suppose that nis even and non-negative, then we have lim ν→n[Pν(x)]2−[Pν(−x)]2 sin(νπ)=2Pn(x) lim ν→nPν(x)−Pν(−x) sin(νπ)=2Pn(x) lim ν→ncos(νπ)Pν(x)−Pν(−x) sin(νπ) =4 πPn(x) lim ν→nQν(x)=4 πPn(x)Qn(x)=4 πPn(−x)Qn(x). If we start from an odd and positive ninstead, we will end up with lim ν→n[Pν(x)]2−[Pν(−x)]2 sin(νπ)=2Pn(x) lim ν→nPν(x)+Pν(−x) sin(νπ)=−2Pn(x) lim ν→ncos(νπ)Pν(x)−Pν(−x) sin(νπ) = −4 πPn(x) limν→nQν(x)=−4 πPn(x)Qn(x)=4 πPn(−x)Qn(x). 26 Hence, we have proved P/integraldisplay 1 −12Pn(ξ)Pn(−ξ) π(x−ξ)dξ=P/integraldisplay 1 −12P−n−1(ξ)P−n−1(−ξ) π(x−ξ)dξ=(−1)n4 πPn(x)Qn(x)=4 πPn(−x)Qn(x),n∈Z≥0, which is equivalent to Eq. 58. In fact, Eq. 58 is not particula rly surprising. It is just a special case of the stronger statement that (cf. [Ref. 27, Eqs. 11.280 and 11.281] or [Ref . 33, Table 1.12A, Eq. 12A.27]) P/integraldisplay 1 −1Pn(ξ)p(ξ) 2(x−ξ)dξ=Qn(x)p(x),degp(x)≤n, (61) where p(x) is any polynomial whose degree does not exceed n. /squaresolid We follow up with some additional examples involving the pro duct of four elliptic integrals in the integrands. Corollary 4.3 (Another Application of Tricomi Pairing) We have an integral identity /integraldisplay 1 −1x[Pν(x)]3Pν(−x)dx=sin(2 νπ)cos(νπ) (2ν+1)2π,ν∈C/integerdivide{−1/2}, (62) which leads to −π 2=/integraldisplay 1 −1x[P−1/2(x)]3P−1/2(−x)dx =32 π4/integraldisplay 1 0(1−2t)[K(/radicallow t)]3K(/radicallow 1−t)dt=−32 π4/integraldisplay 1 0(1−2t)[K(/radicallow 1−t)]3K(/radicallow t)dt, (63) −9/radicallow 3 4π=/integraldisplay 1 −1x[P−1/3(x)]3P−1/3(−x)dx =216/radicallow 3π4/integraldisplay 1 0(1−p2)p(2+p) 1+2p/bracketleftbigg 1−227p2(1+p)2 4(1+p+p2)3/bracketrightbigg/bracketleftBigg K/parenleftBigg/radicalBigg p3(2+p) 1+2p/parenrightBigg/bracketrightBigg3 K/parenleftBigg/radicalBigg 1−p3(2+p) 1+2p/parenrightBigg dp = −72/radicallow 3π4/integraldisplay 1 0(1−p2)p(2+p) 1+2p/bracketleftbigg 1−227p2(1+p)2 4(1+p+p2)3/bracketrightbigg/bracketleftBigg K/parenleftBigg/radicalBigg 1−p3(2+p) 1+2p/parenrightBigg/bracketrightBigg3 K/parenleftBigg/radicalBigg p3(2+p) 1+2p/parenrightBigg dp, (64) −2/radicallow 2 π=/integraldisplay 1 −1x[P−1/4(x)]3P−1/4(−x)dx =32/radicallow 2 π4/integraldisplay 1 01−2u (1+/radicallowu)2/bracketleftBigg K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg3 K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg du = −64/radicallow 2 π4/integraldisplay 1 01−2u (1+/radicallowu)2/bracketleftBigg K/parenleftBigg/radicalBigg 1−/radicallow u 1+/radicallow u/parenrightBigg/bracketrightBigg3 K/parenleftBigg/radicalBigg 2/radicallow u 1+/radicallow u/parenrightBigg du, (65) −27 16π=/integraldisplay 1 −1x[P−1/6(x)]3P−1/6(−x)dx =54 π4/integraldisplay 1 0t(1−t)(1+t)(2−t)(1−2t) (1−t+t2)3[K(/radicallow t)]3K(/radicallow 1−t)dt = −54 π4/integraldisplay 1 0t(1−t)(1+t)(2−t)(1−2t) (1−t+t2)3[K(/radicallow 1−t)]3K(/radicallow t)dt. (66) Proof For integer degrees ν∈Z, the left-hand side of Eq. 62 represents the integration of a n odd function over the interval [ −1,1], hence vanishing. This is consistent with the correspond ing behavior on the right-hand side. We now turn our attention to the scenarios where ν∈C/integerdivide(Z∪{−1/2}). From the Tricomi transform formula in Eq. 57, one may readily deduce the following relation P/integraldisplay 1 −12ξPν(ξ)Pν(−ξ) π(x−ξ)dξ=P/integraldisplay 1 −12[x−(x−ξ)]Pν(ξ)Pν(−ξ) π(x−ξ)dξ=x[Pν(x)]2−[Pν(−x)]2 sin(νπ)−2 π/integraldisplay 1 −1Pν(ξ)Pν(−ξ)dξ. Here, according to Eq. 19 (0,ν), we have T0,ν=/integraldisplay 1 −1Pν(ξ)Pν(−ξ)dξ=2cos( νπ) 2ν+1, 27 so we may combine the formula P/integraldisplay 1 −12ξPν(ξ)Pν(−ξ) π(x−ξ)dξ=x[Pν(x)]2−[Pν(−x)]2 sin(νπ)−4cos( νπ) (2ν+1)π with Eq. 57 for the implementation of the following Tricomi p airing: /integraldisplay 1 −1xPν(x)Pν(−x){[Pν(x)]2−[Pν(−x)]2}dx=/integraldisplay 1 −1xPν(x)Pν(−x)/bracketleftbigg P/integraldisplay 1 −12Pν(ξ)Pν(−ξ)sin(νπ) π(x−ξ)dξ/bracketrightbigg dx = −/integraldisplay 1 −1/bracketleftbigg P/integraldisplay 1 −12ξPν(ξ)Pν(−ξ)sin(νπ) π(x−ξ)dξ/bracketrightbigg Pν(x)Pν(−x)dx = −/integraldisplay 1 −1xPν(x)Pν(−x){[Pν(x)]2−[Pν(−x)]2}dx+2sin(2 νπ) (2ν+1)π/integraldisplay 1 −1Pν(x)Pν(−x)dx. After rearrangement, we obtain 4/integraldisplay 1 −1x[Pν(x)]3Pν(−x)dx=2/integraldisplay 1 −1xPν(x)Pν(−x){[Pν(x)]2−[Pν(−x)]2}dx=2sin(2 νπ) (2ν+1)π/integraldisplay 1 −1Pν(x)Pν(−x)dx, which entails the claimed identity in Eq. 62. The left-hand side of Eq. 62 extends to be a continuous functi on in ν∈C. Taking the ν→−1/2 limit, one arrives at Eq. 63. The special cases ν=−1/3,−1/4,−1/6 correspond to Eqs. 64, 65 and 66. /squaresolid A key step in the proof of Proposition 4.1 hinges on the identi ty /integraldisplay 1 0/bracketleftBig K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig3 dk=3/integraldisplay 1 0[K(k)]2K/parenleftBig/radicalbig 1−k2/parenrightBig dk,i.e./integraldisplay 1 −1[P−1/2(x)]3 /radicallow 1+xdx=3/integraldisplay 1 −1P−1/2(x)[P−1/2(−x)]2 /radicallow 1+xdx. This result is actually just a special case within a family of identities satisﬁed by multiple elliptic integrals involv ing the product of three complete elliptic integrals (of the ﬁrs t and second kinds). To ﬂesh out, for any integer n∈Z, we have P/integraldisplay 1 −1P(2n+1)/2(ξ)/radicalbig 1+ξdξ 2(x−ξ)=Q(2n+1)/2(x)/radicallow 1+x≡(−1)n+1π 2P(2n+1)/2(−x)/radicallow 1+x,∀x∈(−1,1), (67) which generalizes Corollary 3.2(b), and Eq. 67 entails /integraldisplay 1 −1[P(2n+1)/2(x)]3 /radicallow 1+xdx=3/integraldisplay 1 −1P(2n+1)/2(x)[P(2n+1)/2(−x)]2 /radicallow 1+xdx (68) after Tricomi pairing with Eq. 57. In particular, for n=0, Eq. 68 specializes to /integraldisplay 1 0/bracketleftBig 2E/parenleftBig/radicalbig 1−k2/parenrightBig −K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig3 dk=3/integraldisplay 1 0[2E(k)−K(k)]2/bracketleftBig 2E/parenleftBig/radicalbig 1−k2/parenrightBig −K/parenleftBig/radicalbig 1−k2/parenrightBig/bracketrightBig dk. The identity stated in Eq. 67 is the Tricomi transform of P(2n+1)/2(x)//radicallow 1+x≡P−(2n+3)/2(x)//radicallow 1+x,n∈Z, which ex- tends the Neumann integral representation of Legendre func tions Qℓforℓ∈Z≥0(Eq. 60). In the next proposition, we shall prove Eq. 67 in an even broader context, which in turn, a lso gives rise to an independent veriﬁcation of Eq. 57 in Proposition 4.2. Proposition 4.4 (Generalized Neumann Integrals) Forn∈Z≥0andRe(ν−n)>−1, one has P/integraldisplay 1 −1(1+ξ)ν−nPν(ξ)dξ 2(x−ξ)=(1+x)ν−nQν(x),−1<x<1. (69) Proof We note that there is a standard moment formula for Legendre f unctions [Ref. 13, item 7.127]: /integraldisplay 1 −1(1+x)σPν(x)dx=2σ+1[Γ(σ+1)]2 Γ(σ+ν+2)Γ(1+σ−ν),Reσ>−1. (70) One can verify Eq. 70 by termwise integration over the Taylor series expansion for Pν(x)=2F1/parenleftbig−ν,ν+1 1/vextendsingle/vextendsingle1−x 2/parenrightbig with respect to (1−x)/2 and a reduction of the generalized hypergeometric serie s3F2. 28 We now consider the Mellin inversion of Eq. 70: 1 2πi/integraldisplay c+i∞ c−i∞[Γ(s)]22s Γ(s+ν+1)Γ(s−ν)ds (1+ξ)s=/braceleftBigg Pν(ξ),−1<ξ<1 0, ξ>1(71) where c>0 and the integration/integraltext c+i∞ c−i∞:=limT→+∞/integraltext c+iT c−iTis carried out along a vertical line Re s=c. To facilitate analysis, we momentarily assume that Re( ν−n)> −1 2and pick c=Re(ν−n)+1 2, so that c>0 and Re(ν−n−c)=−1 2. For−1<x<1, one may enlist Eq. 71 to compute P/integraldisplay 1 −1(1+ξ)ν−nPν(ξ)dξ 2(x−ξ)=1 2πi/integraldisplay Re(ν−n)+1 2+i∞ Re(ν−n)+1 2−i∞[Γ(s)]22s Γ(s+ν+1)Γ(s−ν)/bracketleftbigg lim ε→0+/integraldisplay ∞ −1/parenleftbigg1 x−ξ+iε+1 x−ξ−iε/parenrightbigg(1+ξ)ν−n−sdξ 4/bracketrightbigg ds =(1+x)ν−n 4i/integraldisplay Re(ν−n)+1 2+i∞ Re(ν−n)+1 2−i∞[Γ(s)]22scot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)ds (1+x)s, (72) where the integration over ξ∈(−1,∞) is an elementary exercise in complex analysis. From Eq. 72, it is straightforward to verify the Legendre differential equation: d dx/bracketleftbigg (1−x2)d dxP/integraldisplay 1 −1(1+ξ)ν−nPν(ξ)dξ 2(1+x)ν−n(x−ξ)/bracketrightbigg +ν(ν+1)P/integraldisplay 1 −1(1+ξ)ν−nPν(ξ)dξ 2(1+x)ν−n(x−ξ) =1 4i/braceleftBigg/integraldisplay Re(ν−n)+1 2+i∞ Re(ν−n)+1 2−i∞[Γ(s)]22scot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)2s2ds (1+x)s+1−/integraldisplay Re(ν−n)+1 2+i∞ Re(ν−n)+1 2−i∞[Γ(s)]22scot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)(s+ν)(s−ν−1)ds (1+x)s/bracerightBigg =1 4i/parenleftBigg/integraldisplay Re(ν−n)+1 2+i∞ Re(ν−n)+1 2−i∞−/integraldisplay Re(ν−n)−1 2+i∞ Re(ν−n)−1 2−i∞/parenrightBigg [Γ(s)]22scot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)2s2ds (1+x)s+1=0 by noting that s=0 and s=ν−nare both removable singularities for the integrand in the la st line. Thus, we are sure that the left-hand side of Eq. 69 can be written as (1 +x)ν−n[aν,nPν(x)+bν,nQν(x)] for some coefﬁcients aν,nand bν,n, provided that Re( ν−n)>−1 2. Instead of directly coping with aν,nandbν,nfor generic νandn, we impose a temporary constraint that n=0,−1 2< ν<0. Again, by Eq. 72, we can compute two moment integrals /integraldisplay 1 −1[aν,0Pν(x)+bν,0Qν(x)]dx=1 2πi/integraldisplay ν+1 2+i∞ ν+1 2−i∞[Γ(s)]2πcot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)ds 1−s =cos(νπ) ν(ν+1)+∞/summationdisplay m=1[Γ(m+ν)]2 (1−m−ν)Γ(m)Γ(m+2ν+1)=−2 ν(ν+1)sin2νπ 2; /integraldisplay 1 −1[aν,0Pν(x)+bν,0Qν(x)](1+x)dx=1 2πi/integraldisplay ν+1 2+i∞ ν+1 2−i∞[Γ(s)]2πcot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)2ds 2−s = −2cos( νπ) (ν−1)ν(ν+1)(ν+2)+∞/summationdisplay m=12[Γ(m+ν)]2 (2−m−ν)Γ(m)Γ(m+2ν+1)=−2[ν2+ν+cos(πν)−1] (ν−1)ν(ν+1)(ν+2) by closing the contours to the right. These two moment integr als reveal that aν,0=0,bν,0=1 for−1 2<ν<0. For any ﬁxed x∈(−1,1), the integral /integraldisplay Reν+1 2+i∞ Reν+1 2−i∞[Γ(s)]22scot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)ds 4i(1+x)s=1 (1+x)νP/integraldisplay 1 −1(1+ξ)νPν(ξ)dξ 2(x−ξ) is analytic for Re ν>−1 2and is equal to Qν(x) for−1 2<ν<0, so it can be identiﬁed with Qν(x) for Re ν>−1 2. Moreover, so long as Re( ν−n)>−1 2for some n∈Z≥0, we have /parenleftBigg/integraldisplay Re(ν−n)+1 2+i∞ Re(ν−n)+1 2−i∞−/integraldisplay Reν+1 2+i∞ Reν+1 2−i∞/parenrightBigg [Γ(s)]22scot(νπ−sπ) Γ(s+ν+1)Γ(s−ν)ds (1+x)s=0, as the ratio cot( νπ−sπ)/Γ(s−ν) remains bounded at all the removable singularities enclos ed in the contour. This allows us to simplify Eq. 72 into Eq. 69 for Re( ν−n)>−1 2,n∈Z≥0. By analytic continuation in ν, one can extend the applicability to Re( ν−n)>−1,n∈Z≥0. /squaresolid 29 Remark One may well recognize that Eq. 69 incorporates the classica l result in Eq. 61 as a special case: P/integraldisplay 1 −1Pn(ξ)p(ξ) 2(x−ξ)dξ=Qn(x)p(x),degp(x)≤n∈Z≥0. To deduce Eq. 57 from Eq. 69, we need the Hardy-Poincaré-Bert rand formula (see [Ref. 31, §4.3, Eq. 4] or [Ref. 27, Eq. 11.52]): /hatwiderT[f(/hatwiderTg)+g(/hatwiderTf)]=(/hatwiderTf)(/hatwiderTg)−f g, (73) which applies to the scenarios where f∈Lp(−1,1),p>1;g∈Lq(−1,1),q>1 and1 p+1 q<1. By Eqs. 69 and 70, we have the following identities valid for −1<ν<0: (1+x)−ν−1Q−ν−1(x)=π 2P/integraldisplay 1 −1(1+ξ)−ν−1P−ν−1(ξ)dξ π(x−ξ), (74) (1+x)ν+1Qν(x)−2ν[Γ(ν+1)]2 Γ(2ν+2)=π 2P/integraldisplay 1 −1(1+ξ)ν+1Pν(ξ)dξ π(x−ξ). (75) Taking the last pair of equations (Eqs. 74 and 75) as inputs fo r the Hardy-Poincaré-Bertrand formula (Eq. 73), we obtain the following identity for −1<ν<0 and−1<x<1: π 2P/integraldisplay 1 −1Pν(ξ)[Qν(ξ)+Q−ν−1(ξ)]dξ π(x−ξ)=Qν(x)Q−ν−1(x). According to the relation between QνandPν(Eq. 3), the last equation reduces to the Tricomi transform o fPν(ξ)Pν(−ξ) (Eq. 57) upon analytic continuation in ν. /square 5 Discussion and Outlook In our evaluations of some generalized Clebsch-Gordan inte grals (Eqs. 25-28), the expressions involving Euler’s gamma function actually correspond to certain special valu es of complete elliptic integrals as well. For example, the knowledge of K(1//radicallow 2)=[Γ(1 4)]2/(4/radicallowπ) [Ref. 13, item 8.129.1] allows us to recast the formula [ Γ(1 4)]8/(128 π2)=/integraltext 1 0[K(/radicallow 1−k2)]3dkinto the following form: 2/bracketleftBigg/integraldisplay 1 0ds /radicallow 1−s2/radicalbig 1−(s2/2)/bracketrightBigg4 =/integraldisplay 1 0/bracketleftBigg/integraldisplay 1 0dt /radicallow 1−t2/radicalbig 1−(1−k2)t2/bracketrightBigg3 dk. (76) Here, both sides are integrations of algebraic functions ov er algebraic domains, which qualify them as members in the “ring of periods” deﬁned by Kontsevich and Zagier [1]. It is g enerally believed that identities for periods can be proved by “algebraic means” [1], namely, relying on nothing else th an additivity of the integral, algebraic change of variable s, and the Newton-Leibniz-Stokes formula. However, the analy tic proof we produced for the generalized Clebsch-Gordan integrals does not fall into such a category: we have invoked Bessel functions, which are “exponential periods” [1]. We would like to see purely algebraic evaluations of the genera lized Clebsch-Gordan integrals when the choice of degree νleads to special values of complete elliptic integrals. In p articular, it could be interesting to search for potential connections to high degree modular equations and the Chowla -Selberg theory. After communicating the ﬁrst version of this manuscript to P rof. Jonathan M. Borwein in January 2013, I was sent a preview of a forthcoming book [34] that contains a sketched proof for Eq. 76 using lattice sums and modular forms, totally independent of the methods presented in my work. Lat er this January, James G. Wan also wrote me about his plan to give a fuller account for the proofs of his own conject ures in a joint work with Rogers and Zucker, currently available as [35]. Some new integrals in [35] have inspired m e to compose a short sequel [36] to the current work, in which there are further applications of the Hardy-Poinca ré-Bertrand formula (Eq. 73) and the Tricomi transform of (1+ξ)ν−nPν(ξ),n∈Z≥0,Re(ν−n)> −1 (Eq. 69), as well as an extension of the Hansen-Heine scalin g analysis in Proposition 2.1 to the computations of other multiple ellip tic integrals. The spherical methods in this work (Legendre functions and s pherical rotations) enable us to handle a large variety of multiple elliptic integrals, but they are by no means a cur e-all. The methods of Bessel moments [2, 4, 5], geometric transformations of Watson type [3], hypergeometric summat ions [6] still play fundamental rôles in our quantitative understandings for integrals over elliptic integrals. 30 With a synthesis of various techniques, it is sometimes poss ible to handle integrals over the product of more than three Legendre functions of the same degree ν∈C, such as /integraldisplay 1 −1x[Pν(x)]4dx=limz→ν2sin4(πz)[ψ(2)(z+1)+ψ(2)(−z)+28ζ(3)] (2z+1)2π4,where ψ(2)(z) :=d3 dz3logΓ(z). (77) One may wish to compare Eq. 77 to the integrals/integraltext 1 −1x[Pν(x)]3Pν(−x)dx,ν∈C(Eq. 62) that reduce to elementary functions. Clearly, special cases of Eq. 77 bring us some int eresting evaluations of multiple elliptic integrals. For example, we have the following integral representations fo r Apéry’s constant ζ(3)=/summationtext∞ n=1n−3: ζ(3)=−π4 243/integraldisplay 1 −1x[P−1/3(x)]4dx=−π4 168/integraldisplay 1 −1x[P−1/4(x)]4dx=−2π4 189/integraldisplay 1 −1x[P−1/6(x)]4dx as well as a critical scenario involving ζ(5)=/summationtext∞ n=1n−5: ζ(5)=−π4 372/integraldisplay 1 −1x[P−1/2(x)]4dx=8 93/integraldisplay 1 0(2t−1)[K(/radicallow t)]4dt. Acknowledgements The author thanks two anonymous referees for their thoughtf ul comments that helped improve the presentation of this paper. This work was partly support ed by the Applied Mathematics Program within the Department of Energy (DOE) Ofﬁce of Advanced Scientiﬁc Comp uting Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4). Th e author thanks Prof. Weinan E (Princeton University) for his encouragements. References [1] Maxim Kontsevich and Don Zagier. Periods. 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