IntegralTransformsSpecFunct.v25y2014p312

Integral Transforms and Special Functions , 2014 V ol. 25, No. 4, 312–317, http: //dx.doi.org /10.1080 /10652469.2013.842235 A Dirac delta-type orthogonality relation for the on-the-cut generalized associated Legendre functions of the first kind with imaginary second upper indices Radosław Szmytkowski∗and Sebastian Bielski Atomic Physics Division, Department of Atomic, Molecular and Optical Physics, Faculty of Applied Physics and Mathematics, Gda´ nsk University of Technology, Narutowicza 11 /12, PL 80–233 Gda´ nsk, Poland (Received 4 May 2013; final version received 4 September 2013 ) The integral/integraltext1 −1dx(1+x)−1Pλ,iμ ν(x)Pλ,iμ/prime ν(x), involving the on-the-cut generalized associated Legendre functions of the first kind with ν∈C,R eλ< 1o rλ∈N+,μ,μ/prime∈R, is evaluated in a closed form. It is found to be proportional to 2iμ/2δ(μ−μ/prime)+2−iμ/2δ(μ+μ/prime), where δ(μ∓μ/prime)is the Dirac delta distribution. Keywords: special functions; generalized associated Legendre functions; orthogonal functions; the Dirac delta distribution MSC 2010 : 33C05; 33C47 1. Introduction Among various types of index transforms,[ 1] an important class is formed by those with kernels involving the Legendre functions or the associated Legendre functions. Examples encompass thewell-known Mehler–Fock transform [ 2,3] (see also [ 1, chap. 3]) and its various generalizations (see, e.g. [ 4–6]). In Ref. [ 7], Götze considered a particular generalization of the Mehler–Fock transform with an integral kernel involving the generalized associated Legendre function of the first kind P λ,μ ν(x), introduced by Kuipers and Meulenbeld [ 8] (see also [ 9]). From the results presented in Ref. [ 7], one may deduce the following Dirac delta-type orthogonality relation: /integraldisplay∞ 1dxPλ,μ −1/2+iκ(x)Pλ,μ −1/2+iκ/prime(x)=2μ−λ+1π2[δ(κ−κ/prime)+δ(κ+κ/prime)] κsinh(2πκ) ×/Gamma1−1((1−λ+μ)/2+iκ)/Gamma1−1((1−λ+μ)/2−iκ) ×/Gamma1−1((1−λ−μ)/2+iκ)/Gamma1−1((1−λ−μ)/2−iκ) (κ,κ/prime∈R;R eλ< 1o rλ∈N+;μ∈C), (1.1) where δ(κ∓κ/prime)is the Dirac delta distribution. ∗Corresponding author. Email: [email protected] © 2013 Taylor & Francis Downloaded by [Radosaw Szmytkowski] at 08:13 20 March 2014 Integral Transforms and Special Functions 313 It is the purpose of the present note to provide another Dirac delta-type orthogonality relation obeyed by the generalized associated Legendre functions of the first kind. Specifically, we shallprove that on the cut −1/lessorequalslantx/lessorequalslant1 these functions satisfy /integraldisplay 1 −1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x=2iμ/2−λ+2π2[2iμ/2δ(μ−μ/prime)+2−iμ/2δ(μ+μ/prime)] μsinh(πμ) ×/Gamma1−1(ν+1−(λ+iμ)/2)/Gamma1−1(ν+1−(λ−iμ)/2) ×/Gamma1−1(−ν−(λ+iμ)/2)/Gamma1−1(−ν−(λ−iμ)/2) (ν∈C;R eλ< 1o rλ∈N+;μ,μ/prime∈R). (1.2) The method we shall use below to arrive at the relation in Equation ( 1.2) is analogous to the one we have employed in our previous works [ 10–12] on the Dirac delta-type orthogonality relations for some other special functions of mathematical physics. 2. Summary of relevant properties of the generalized associated Legendre function of the first kind In the complex plane cut along the real axis from z=− ∞ toz=1, the generalized associ- ated Legendre function of the first kind may be defined in terms of the Gauss’ hypergeometricfunction as P λ,μ ν(z)=1 /Gamma1(1−λ)(z+1)μ/2 (z−1)λ/22F1/parenleftbigg ν+1−λ−μ 2,−ν−λ−μ 2;1−λ;1−z 2/parenrightbigg . (2.1) The function ( 2.1) obeys the differential identity d dz(1−z2)dPλ,μ ν(z) dz+/bracketleftbigg ν(ν+1)−λ2 2(1−z)−μ2 2(1+z)/bracketrightbigg Pλ,μ ν(z)=0. (2.2) On that part of the cut for which z=x∈[ − 1, 1], we define the on-the-cut function Pλ,μ ν(x)in terms of either of the limits Pλ,μ ν(x±i0)as Pλ,μ ν(x)=e±iπλ/2Pλ,μ ν(x±i0)(−1/lessorequalslantx/lessorequalslant1). (2.3) Hence, with the aid of the relation z−1=e±iπ(1−z)(Imz≷0), (2.4) it follows that Pλ,μ ν(x)=1 /Gamma1(1−λ)(1+x)μ/2 (1−x)λ/22F1/parenleftbigg ν+1−λ−μ 2,−ν−λ−μ 2;1−λ;1−x 2/parenrightbigg (−1/lessorequalslantx/lessorequalslant1). (2.5) The on-the-cut function Pλ,μ ν(x)satisfies the differential identity obtained from Equation ( 2.2) through the formal replacement z→x. Downloaded by [Radosaw Szmytkowski] at 08:13 20 March 2014 314 R. Szmytkowski and S. Bielski In Section 3, we shall rely on asymptotic representations of the on-the-cut function Pλ,μ ν(x) asx→± 1∓0. The representations valid for x→1−0 may be immediately deduced from Equation ( 2.5); these are Pλ,μ ν(x)x→1−0−→2(μ−λ)/2 /Gamma1(1−λ)/parenleftbigg1−x 2/parenrightbigg−λ/2 [1+O(1−x)](λ/negationslash∈N+) (2.6a) and Pλ,μ ν(x)x→1−0−→2(μ−λ)/2 /Gamma1(λ+1)/Gamma1(ν+1+(λ+μ)/2)/Gamma1(−ν+(λ+μ)/2) /Gamma1(ν+1−(λ−μ)/2)/Gamma1(−ν−(λ−μ)/2)/parenleftbigg1−x 2/parenrightbiggλ/2 ×[1+O(1−x)](λ∈N+). (2.6b) The counterpart representation valid for x→− 1+0i s Pλ,μ ν(x)x→− 1+0−→2(μ−λ)/2/Gamma1(−μ) /Gamma1(ν+1−(λ+μ)/2)/Gamma1(−ν−(λ+μ)/2)/parenleftbigg1+x 2/parenrightbiggμ/2 [1+O(1+x)] +2(μ−λ)/2/Gamma1(μ) /Gamma1(ν+1−(λ−μ)/2)/Gamma1(−ν−(λ−μ)/2)/parenleftbigg1+x 2/parenrightbigg−μ/2 [1+O(1+x)], (2.7) as may be inferred from Equation ( 2.5) and from the Gauss’ relation [ 13, Equation (9.131.2)] 2F1(a1,a2;b;z)=/Gamma1(b)/Gamma1(b−a1−a2) /Gamma1(b−a1)/Gamma1(b−a2)2F1(a1,a2;a1+a2−b+1; 1−z) +(1−z)b−a1−a2/Gamma1(b)/Gamma1(a1+a2−b) /Gamma1(a1)/Gamma1(a2) ×2F1(b−a1,b−a2;b−a1−a2+1; 1−z). (2.8) 3. A Dirac delta-type orthogonality relation involving the on-the-cut generalized associated Legendre functions of the first kind with imaginary second upper indices Throughout this section, it will be assumed that −1/lessorequalslantx/lessorequalslant1,ν∈C,μ,μ/prime∈R. (3.1) For the time being, we also assume that λ∈C. Later on, some necessary restrictions on λwill be imposed. Consider the two on-the-cut functions Pλ,iμ ν(x)andPλ,iμ/prime ν(x). According to what has been said in Section 2, they satisfy the differential identities d dx(1−x2)dPλ,iμ ν(x) dx+/bracketleftbigg ν(ν+1)−λ2 2(1−x)+μ2 2(1+x)/bracketrightbigg Pλ,iμ ν(x)=0 (3.2) and d dx(1−x2)dPλ,iμ/prime ν(x) dx+/bracketleftbigg ν(ν+1)−λ2 2(1−x)+μ/prime2 2(1+x)/bracketrightbigg Pλ,iμ/prime ν(x)=0. (3.3) Downloaded by [Radosaw Szmytkowski] at 08:13 20 March 2014 Integral Transforms and Special Functions 315 Premultiplying Equation ( 3.2)b yPλ,iμ/prime ν(x), Equation ( 3.3)b yPλ,iμ ν(x), subtracting and integrating the resulting equation over xfrom x1tox2, where −1<x1<x2<1, yields /integraldisplayx2 x1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x =2 μ/prime2−μ2(1−x2)/bracketleftBigg Pλ,iμ/prime ν(x)dPλ,iμ ν(x) dx−Pλ,iμ ν(x)dPλ,iμ/prime ν(x) dx/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex=x2 x=x1. (3.4) It follows from Equations ( 2.6a) and ( 2.6b) that the integral on the left-hand side of Equation ( 3.4) converges as x2→1−0, provided that λis constrained to obey Reλ< 1o r λ∈N+. (3.5) Henceforth, the restrictions ( 3.5) will be assumed to hold. Then, again with the aid of Equa- tions ( 2.6a) and ( 2.6b), one finds that the expression on the right-hand side of Equation ( 3.4) vanishes for x2=1. Hence, it follows that /integraldisplay1 x1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x =2 μ/prime2−μ2(1−x2 1)/bracketleftBigg Pλ,iμ ν(x1)dPλ,iμ/prime ν(x1) dx1−Pλ,iμ/prime ν(x1)dPλ,iμ ν(x1) dx1/bracketrightBigg . (3.6) To proceed further, we observe that Equation ( 2.7) implies the asymptotic formula Pλ,iμ ν(x)x→− 1+0−→ 2(iμ−λ)/2/bracketleftbigg Aλ,μ νexp/parenleftbiggiμ 2ln1+x 2/parenrightbigg +Aλ,−μ νexp/parenleftbigg −iμ 2ln1+x 2/parenrightbigg/bracketrightbigg , (3.7) where Aλ,±μ ν=/Gamma1(∓iμ) /Gamma1(ν+1−(λ±iμ)/2)/Gamma1(−ν−(λ±iμ)/2). (3.8) Making use of Equation ( 3.7) on the right-hand side of Equation ( 3.6) results in /integraldisplay1 −1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x=2i(μ+μ/prime)/2−λ+1 ×lim x→− 1+0/bracketleftbigg (Aλ,μ νAλ,−μ/prime ν+Aλ,−μ νAλ,μ/prime ν)sin(−((μ−μ/prime)/2)ln((1+x)/2)) μ−μ/prime +(Aλ,μ νAλ,μ/prime ν+Aλ,−μ νAλ,−μ/prime ν)sin(−((μ+μ/prime)/2)ln((1+x)/2)) μ+μ/prime +iAλ,μ νAλ,−μ/prime ν−Aλ,−μ νAλ,μ/prime ν μ−μ/primecos/parenleftbiggμ−μ/prime 2ln1+x 2/parenrightbigg +iAλ,μ νAλ,μ/prime ν−Aλ,−μ νAλ,−μ/prime ν μ+μ/primecos/parenleftbiggμ+μ/prime 2ln1+x 2/parenrightbigg/bracketrightBigg . (3.9) To take the limit on the right-hand side of Equation ( 3.9), we observe that in the distributional sense it holds that (cf., e.g. [ 14, p. 35] or [ 15, vol. I, p. 43]) lim a→∞sinaη πη=1 2πlim a→∞/integraldisplaya −adξeiξη=δ(η), (3.10a) Downloaded by [Radosaw Szmytkowski] at 08:13 20 March 2014 316 R. Szmytkowski and S. Bielski where δ(η) is the Dirac delta distribution, and lim a→∞cosaη=0 (3.10b) (the latter identity is a corollary from the Riemann–Lebesgue lemma). Since for x→− 1+0 one has ln[(1+x)/2]→− ∞ , application of Equations ( 3.10a ) and ( 3.10b ) transforms Equation ( 3.9) into the distributional identity /integraldisplay1 −1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x=2i(μ+μ/prime)/2−λ+1π[(Aλ,μ νAλ,−μ/prime ν+Aλ,−μ νAλ,μ/prime ν)δ(μ−μ/prime) +(Aλ,μ νAλ,μ/prime ν+Aλ,−μ νAλ,−μ/prime ν)δ(μ+μ/prime)]. (3.11) However, it follows from elementary properties of the Dirac delta that Aλ,μ/prime νδ(μ∓μ/prime)=Aλ,±μ νδ(μ∓μ/prime). (3.12) With this, after employing Equation ( 3.8) and using the well-known relation |/Gamma1(iμ)|=/radicalbiggπ μsinh(πμ)(μ∈R), (3.13) Equation ( 3.11) is transformed into the Dirac delta-type orthogonality relation /integraldisplay1 −1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x=2iμ/2−λ+2π2[2iμ/2δ(μ−μ/prime)+2−iμ/2δ(μ+μ/prime)] μsinh(πμ) ×/Gamma1−1(ν+1−(λ+iμ)/2)/Gamma1−1(ν+1−(λ−iμ)/2) ×/Gamma1−1(−ν−(λ+iμ)/2)/Gamma1−1(−ν−(λ−iμ)/2) (ν∈C;R eλ< 1o rλ∈N+;μ,μ/prime∈R), (3.14) that has been already announced in the introduction to be the central result of this work. Concluding, we observe that the relation in Equation ( 3.14) may be split into /integraldisplay1 −1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x=2iμ−λ+2π2δ(μ−μ/prime) μsinh(πμ) ×/Gamma1−1(ν+1−(λ+iμ)/2)/Gamma1−1(ν+1−(λ−iμ)/2) ×/Gamma1−1(−ν−(λ+iμ)/2)/Gamma1−1(−ν−(λ−iμ)/2) (ν∈C;R eλ< 1o rλ∈N+;μ,μ/prime∈R; signμ=signμ/prime) (3.15) and /integraldisplay1 −1dxPλ,iμ ν(x)Pλ,iμ/prime ν(x) 1+x=2−λ+2π2δ(μ+μ/prime) μsinh(πμ) ×/Gamma1−1(ν+1−(λ+iμ)/2)/Gamma1−1(ν+1−(λ−iμ)/2) ×/Gamma1−1(−ν−(λ+iμ)/2)/Gamma1−1(−ν−(λ−iμ)/2) (ν∈C;R eλ< 1o rλ∈N+;μ,μ/prime∈R; signμ/negationslash=signμ/prime). (3.16) It is easy to see that the two relations are not independent as Equation ( 3.16) may be deduced from Equation ( 3.15) after one replaces in the latter μby−μand uses the identity [ 9, Equation (4.2)] Pλ,−iμ ν(x)=2−iμPλ,iμ ν(x). (3.17) Downloaded by [Radosaw Szmytkowski] at 08:13 20 March 2014 Integral Transforms and Special Functions 317 References [1] Yakubovich SB. Index transforms. Singapore: World Scientific; 1996. 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