kalia dual integral

Internat. J.Math. 6Math, Sei. 257 DUAL INTEGRAL EQUATIONS WITH FOX’S H-FUNCTION KERNEL RN.KALIA 1. srmsonoctton forsee to.x61). an Jacoxconoen =eG, xelas a. 258 Ra, KALIA 2. THE SOLUTION. tieshall find the formal solution ofthedual integral equations given by Fa (apo)Jcoyale|eeay=e) (<x<D ey% ; (bqr8q) > (apsap)Jfowin[»|?Jo=0 > 2.2)4 Ps (aot) where mn,p and qare integers with 1=m <q, 0s 0p, p<4, and @=0,where re reee eyinel 1 ml and Gassag) VeeTPOy8ys) TPC-aytays) ametleg ft 41 _____ 8as (2.4) 1 a ee a Pvie Trae ytays) MT(ay-ays)gent jentl ».-1 4 os Weapply the formal manipulations given byTitchmarsh to (2.1) and (2.2). Tf Parseval's formula isapplied tothe left-hand sides of (2.1) and (2.2), the results .oye 4Peosteytaey-99 aodF(s) det___-_______- wee T_T(by-8y-0844838)seme TP(-ay-aj-a0 tags)3 ET an=60 @exen es) eres? and ac 1Feost8y-839)afeo +———_- Wie -b5-8, 48,8seater eyt8y0 Tr(a,ene ec) a lasso, Gp. 2.61 Tayta.naysper Oss? DUAL INTEGRAL EQUATIONS UITH FOX'S H-FUNCTTON KERNEL 259 The substitution of 1 (. )HTr ) TorUsby-6y88) MT(atatao =ji5 = jy 33 roedemeFFgem FSacey an Tr(byta8j#8,-8)8) TlT(L-a,-as4a58)genOSegPg8gTPOragragtng into the last two equations gives cs t(mb, >eT Ub 8488) Ay{wot i 7 Keto oT rab a8, -8,48,8) gmt SSS ae ee vat xT asec), reed, 2.8) Tr(-aj-ayta,s) Pn kets TT(eyty-8ys) azJrs—--- ai= Wie rym 48-80) iH( )=: J35 5 = +Hatt——____ slgs=0+eb . 2.9 1 rlaytay-a,e) gen 999 Multiplying (2.8) byx"” ,where Re(s-w) >0andintegrating over (0,1) ,we obtain iSATC-b, )He Tr y-8, 4858 a ett SS : ti 7 issheetCby8y8y*8,)) Tr(-aj-a,-na stays) :Crane ieaeMeXs) ow Tr(-aj-a,ta,s) a gd =Ee w+ 1) ew <w . 2.10) Moving the Line of integration from Reatk to Re sek’ <Rew 8D (agta tna,-1)andassumingReLE CysGjemttcooya) andReSE yeetsy ) y 260 ans RALIA we obtain ¢ ° 1Ree Eamtayaiyey Ereapeyeyy iteFranti Braacatan Hordby-8,48,6) BPeay-ay aa 9) =etetem ge ee ean Eyrerseasy tty Hecate The integral occurring on the left-hand side of this equation isaregular function 4 2 atte att FEr-ayaoestayn 1fweantune suitable conditions at infinity, ve have ete Forcsbyo8y-8)48)8) TrCleajnaytaye)afaeaea caso|a57° ieghee as) rays? (eww) ei Similarly, multiplying (2.9) by0” ,Re(s-w) <0, and integrating over (1,"), we obtain wets TTrCeytb,-@)8) TMraytaytoy-ays) Bef ht ee, ce, cnEteRPreseyescaey Ereytaayn Tretoy eye) TC » shOates ey?sheOM” DUAL INTEGRAL EQUATIONS WH. FOX'S HEFINCTTON KERNEL 261 wate 1 1) .tf Ware ewok gaswee Moving the Line ofintegration from Restk! toReork ,vehave ese 1 1)af aw even eae wee te follows from (2.12) and (2.35) that wie rasby-o8 8,48 Br(-aj-a,4a,8)1 Serie eeieMeeae eeae ie Eran sy Freon) HOHE ay aan XfMetiin's inversion formla 4sapplied to (2.7) «then t+HTU-b, )a( )te H raseesteye) FE rayteytan cays ekebieareMeee os cone fe eeIcova aa JoeT1o408,48)-8,8) 1TUnaycaytays BieTregaigtayey Beaearesnye Equations (2.17) and (2.18) give asolution to (2.1) and (2.2) « The cases ofMetjer"s C-function and lover transcendents follov inaperspicuous sanner onparticularizing the paraneter inthe H-funetion. The Aéencity aya fatYaa, Aadaae "| oy8D (oye8498) allows the absorption ofany pover of x inthe H-funetion. Therefore, we could have multiplied thekernels of (2.1) byx",put x"e(x) =g(x) andsolved thepair thus obtained without any Loss ofgenerality. 262 RAN, KALTA REFERENCES 1. SNEDDON, T.N. Mixed Boundary Value Problems inPotential Theory, North-Holland (1968). 2. JOHNSON, B.C. Integral Equations involving Special Functions. Ph.D. Thesis, Oregon State University (1964). 3. TITCHMARSH, E.C. Introduction tothe Theory of Fourier Integrals, 2nd Ed. Oxford (1948).” 4, KESARWANI, R.N. Fractional integration and certain dual integral equations, Math. Z.98 (1967) 83-88. 5. KESARWANI, R.N. Correction tofractional integration and dual integral equations, Math. Z.107 (1968) 82. 6. BUSCHMAN, R.G. Fractional integration, Math. Japan 9(1964) 99-106. 7. SAXENA, R.K, Aformal solution ofcertain dual integral equations involving H-function, Proc. Camb. Phil. Soc. 63 (1967) 171-178. 8. SAKENA, R.K. On the formal solution ofdual integral equations, Proc. Amer. Math. Soc. 18 (1967) 1-8.