On Generating Relations of Some Triple. Hypergeometric Functions

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1 It. Joural of Math. Aalysis, Vol. 5,, o., 5 - O Geeratig Relatios of Some Triple Hypergeometric Fuctios Fadhle B. F. Mohse ad Gamal A. Qashash Departmet of Mathematics, Faculty of Educatio Zigibar Ade Uiversity, Ade, Yeme mfazalmohse@yahoo.com Abstract I this paper we aim at presetig certai geeratig relatios, ivolvig the Exto's fuctios 7 8 ad, where every oe ca be represeted by Laplace itegral formulae. Some of kow results are obtaied as special cases of our mai results. Mathematics Subject Classificatio: 33C5, 33C56 Keywords: Laplace trasforms, three variables hypergeometric fuctios, geeratig relatios.. Itroductio Exto i [] defied ad gave itegral represetatios of some hypergeometric fuctios of three variables, ad deoted them by,.... The defiitios ad itegral represetatios of 7 8 ad, are as follows (cf. []). m p ( α) m+ + p ( β ) p x y z, ;,, ;,, =,. ( γ) ( δ) λ m!! p! ( αβγδλx y z ) ( αβγδλx y z ) mp,, = m p mp,, = m p m p ( α) m+ + p( β ) + p x y z, ;,, ;,, =, (.) ( γ) ( δ) λ m!! p!

2 6 F. B. F. Mohse ad G. A. Qashash ( αβ β γδx y z ) 7 mp,, = m + p ( α) ( ) m p m+ + p β β p x y z,, ;, ;,, =, (.3 ) ( γ) ( δ) m!! p! ( αβ β γδλx y z ) 8 ( α) ( ) m p m+ + p β β p x y z,, ;,, ;,, =, (.) ( γ) ( δ) λ m!! p! ( αβγδλx y z ) mp,, = m p m p ( α) m+ ( β ) + p x y z, ;,, ;,, =,.5 ( γ) ( δ) λ m!! p! mp,, = m p s α ( αβγδλ, ;,, ;,, ) = ; ; Γ ( α ) ( γ ) x y z e s F xs ( ; δ ; ) ( β ; λ ; ), (.6 ) F ys F zs ds s α ( αβγδλ, ;,, ;,, ) = ; ; Γ ( α ) ( γ ) x y z e s F xs Ψ s α 7( αβ,, β; γδ, ;,, ) = ; γ; Γ ( α ) ( βδλ ;, ; ys, zs ) ds, (.7) x y z e s F xs Φ ( β, β ; δ; ys, zs ) ds, (.8) s α 8( αβ,, β; γδλ,, ;,, ) = ; γ; Γ ( α ) x y z e s F xs ( β ; δ; ) ( β ; λ; ), (.9) F ys F zs ds s t α β ( αβγδλ, ;,, ;,, ) = ; ; Γ ( α) Γ ( β) ( γ ) x y z e s t F xs ( ; δ; ) ( ; λ; ), (.) F ys t F zt ds dt

3 Geeratig relatios of triple hypergeometric fuctios 7 where Re ( α ) >, Re( β ) >, fuctios (cf. [], p.58) ad the fuctios i the itegrad are Humbert. Results We have established the followig ew geeratig fuctios ( α +, β; γ, δ, λ; x, y, z ) w =! w α + α + + = =! β ( z ) F, ; γδ, ; x, y, (.) =! ( α +, β; γ, δ, λ;,, ) x y z w = ( x + y =! x + y ( 3) z x y FA α +, β, γ, δ ; λ,γ,δ ;,,,. x + y + x + y + x + y = ( α +, β; γ, δ, λ; x, y, z ) w! = ( x =! x y x z 8 α +, γ, β; δ,γ, λ;,,,.3 (x x + x ( α +, β; γ, δ, λ; x, y, z ) w = ( z )!! z = = x y z α +, λ β; γ, δ, λ;,,,. ( z) ( z) z

4 8 F. B. F. Mohse ad G. A. Qashash ( α +, β; γ, δ, λ; x, y, z ) w = ( x!! x = = x y z FE α +, α +, α +, γ, β, β;γ, δ, λ;,,,.5 x + x + x ( α +, β, β ; γ, δ; x, y, z ) w = ( x!! x 7 = = x y z FG α +, α +, α +, γ, β, β;γ, δ, δ;,,,.6 x + x + x ( α +, β, β ; γ, δ, λ; x, y, z ) w = ( x!! x 8 = = ( 3) x y z FA α +, γ, β, β;γ, δ, λ;,,,.7 x + x + x ( α +, β + ; γ, δ, λ;,, ) = ( +!! x + z x y z w x z = = ( 3) y x z FA α +, β +, γ, λ ; δ,γ,λ ;,,,.8 x + z + x + z + x + z + 3 where F is the Appell's fuctio (cf. []), F A is the three variables Lauricella fuctio (cf,[],p.6), ad F E, FG are Sara's fuctios of three variables (cf.[],p.67). 3. Proofs of results For provig the above results, we eed the followig formulae (cf. [], p.37, 7 ad [3], p.57)

5 Geeratig relatios of triple hypergeometric fuctios 9 x ( i ) F ( a; c; x ) = e F ( c a; c; x ) ; ; x ii F c x = e F c ; c ; x λ λ + iii λ =,,,,3,... = γ α γ { αγλ } ( γ) ( λ) ( γ) ( λ) iv L t F ; ; t =Γ p p, Re >,Re p >,Re > { ν ν } ( ν ) ( ν) ( v) L t =Γ + p, Re >,Re( p) >. μ μ ω σ ( vi ) L { x F( a, b; σx ) F( c, d ; ωy )} =Γ ( μ p F μ +, a, c; b, d ;, p p ( μ) ( p σ) ( p ω) ( p σ ω) Re >, Re,Re,Re >. Where L is the Laplace trasform, ad F is Appell's fuctio. Proof of (.) Let us deote the left had side of (.) by I, usig (.6) w s α + I = e s ( ) ( F ; γ ; x s F ; δ ; y s ) F( β ; λ ; zs) ds Γ +! ( α ) Now, writig the secod of two fuctios F i the itegrad i its series form, ad iterchagig the order of the summatio ad itegral sig which is permissible here, we get p q w x y s ( α + + p+ q ) I = e s F( βλ ; ; zs) ds, ( 3.), p, q= Γ ( α + )( γ) ( δ)! p! q! p q Also usig (iv), we get p q w x y Γ ( α + + p + q) I =,,, ( α )( γ) ( δ)! p! q! z β = Γ + p q p q = p q ( α + ) w x y p+ q ( γ) ( δ)! p! q! ( ), p, q= p q z β, 3.

6 F. B. F. Mohse ad G. A. Qashash Now i (3.), simplified by usig series maipulatio (iii), cosiderig the defiitio of the Appell's fuctio F,we will get the right had side of (.), which complete the Proof of (.). The proof of (.) to (.8) rus i the same way, cosiderig the appropriate itegral represetatio ad Laplace trasform durig the proof.. Special cases Some geeratig relatios, which believed to be ew, ca be established as special cases as follow: I (.), choosig x =, we get the followig relatio. =! ( α +, β; δ, λ;, ) H y z w w α + α + + = =! β ( z ) F, ; δ ; y, (.) I (.), put z =, we get = α + α + +! F, ; γδ, ; x,y w = ( x + y =! x + y x y F α +, γ, δ ;γ,δ ;,,. x + y + x + y I (.3), put y =, we get =! ( α +, β; δ, λ;, ) H x z w x z = + + =! x x + x ( x ) F α, γ, β;γ, λ;,, (.3) From (.) ad (.), we have

7 Geeratig relatios of triple hypergeometric fuctios α + α + + ( β z) F, ; γ, δ ;x, y w! = ( x + y =! x + y ( 3) z x y FA α +, β, γ, δ ; λ,γ,δ ;,,,. x + y + x + y + x + y I (.), put z =, we get w α + α + + F, ; γ, δ ;x!, y = ( x + y =! x + y x y F α +, γ, δ ;γ, δ ;,,.5 x + y + x + y I (.5), put z =, we get H ( α +, β; γ, δ; x, y ) w = ( x!! x = = x y F α +, γ, β;γ, δ;,,.6 x + x I (.6) ad (.7), put y =, we have H ( α +, β ; γ, δ; x, z ) w = ( x!! x = = x z F α +, γ, β;γ, δ;,,.7 x + x

8 F. B. F. Mohse ad G. A. Qashash H ( α +, β ; γ, λ; x, z ) w = ( x!! x = = x z F α +, γ, β;γ, λ;,,.8 x + x I (.8), put x =, we get H ( β +, α + ; λ, δ; z, y ) w = ( z!! z = = y z F α +, β +, λ ; δ,λ ;,,.9 z + z Refereces [] Erdelyi, A., Magus, W., Oberhettiger, F. ad Tricomi, F.G. Tables of Itegral Trasforms, vol.. McGraw-Hill New York, Toroto ad Lodo, 95. [] Exto,H., Multiple Hyper geometric Fuctios ad Applicatios, EllisHorwood Ltd., Chichester,U.K., 976. [3] Prudikov, A.P.,Brychkov,Yu.A., ad Marichev,O.I. (), Itegrals ad Series, Direct Laplace Trasform, Vol..Gordo ad Breach Sciece Publishers, 99. [] Srivastava, H.M. ad Maocha, H.L. (), A Treaties o Geeratig Fuctios, Halsted press, Joh Wiley ad Sos, New York, 98. Received: Jue,

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