Wright Type Hypergeometric Function and Its Properties

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1 Advane in Pure Mahemai h://ddoiorg/4236/am Publihed Online May 23 (h://wwwirorg/journal/am) Wrigh Tye Hyergeomeri union and I Proerie Snehal B Rao Jyoindra C Prajaai 2 Ajay K Shula 3 Dearmen of Alied Mahemai The MS Univeriy of Baroda Vadodara India 2 Dearmen of Mahemaial Siene auly of Alied Siene Charoar Univeriy of Siene and Tehnology Anand India 3 Dearmen of Alied Mahemai and Humeniie SV Naional Iniue of Tehnology Sura India br_mub@yahooom jyoindra8@rediffmailom ajayhula2@rediffmailom Reeived January 7 23; revied ebruary 6 23; aeed Marh 6 23 Coyrigh 23 Snehal B Rao e al Thi i an oen ae arile diribued under he Creaive Common Aribuion iene whih ermi unreried ue diribuion and reroduion in any medium rovided he original wor i roerly ied ABSTRACT e and be omle variable be he Gamma funion and v v for any omle v be he generalied Pohhammer ymbol Wrigh Tye Hyergeomeri union i defined (Virheno e al []) a: a b 2Ra b; ; ; where ; ab ; Rea Reb Re ; b! whih i a dire generaliaion of laial Gau Hyergeomeri union ab ; ; The rinial aim of hi aer i o udy he variou roerie of hi Wrigh ye hyergeomeri funion R ab ; ; ; whih inlude ; differeniaion and inegraion rereenaion in erm of and in erm of Mellin-Barne ye inegral Euler (Bea) ranform alae ranform Mellin ranform Whiaer ranform have alo been obained; along wih i relaionhi wih o H-funion and Wrigh hyergeomeri funion Keyword: Euler Tranform; o H-union; Wrigh Tye Hyergeomeri union; alae Tranform; Mellin Tranform; Whiaer Tranform; Wrigh Hyergeomeri union Inroduion and Preliminarie The Gau Hyergeomeri union i defined [2] a: ab ; ; a b! 2 ; and () The Generalied Hyergeomeri union in a laial ene ha been defined [3] by a a ; a a a a b b b b ; ; b b;! (2) and no denominaor arameer eual o ero or negaive ineger E Wrigh [4] ha furher eended he generaliaion of he hyergeomeri erie in he following form where r and n n n n n n ; n! (3) are real oiive number uh ha r When r and are eual o Euaion (3) differ from he generalied hyergeomeri funion by a onan mulilier only The generalied form of he hyergeomeri funion ha been inveigaed by Doeno [5] Maloviho [6] and one of he eial ae onidered by Doeno [5] a r Coyrigh 23 SiRe

2 336 S B RAO ET A R R a b ; ; ; an b n a b n n n! and i inegral rereenaion ereed a b b a b R where Re b b d n (4) (5) Re Thi i he analogue of Euler formula for he Gau hyergeomeri funion [3] In 2 Virheno e al [] defined he aid Wrigh Tye Hyergeomeri union by aing in (4) a ; ; ; a b b! R R a b ; (6) If hen (3) redue o a Gau hyer- geomeri funion Galue e al [7] and Virheno e al [] inveigaed ome roerie of he funion 2R ab ; ; ; The following well-nown fa have been reared for udying var iou roerie of he funion 2R ab ; ; ; Euler (Bea) ranform (Sneddon [8]): The Euler ranform of he funion f i defined a a b B f : a b f d (7) alae ranform (Sneddon [8]): The alae ranform of he funion f fined a i de- f e f d (8) i de- Mellin ranform (Sneddon [8]): The Mellin ranform of he funion f fined a hen ; d M f f f Re f M f ; d 2πi f (9) () Wrigh generalied hyergeomeri funion (Srivaava and Manoha [9]) denoed by i defined a H A A; B B; i i A i! j j B j A A B B mn where H H-funion [] a a b b 2 Bai Proerie of he union R ab () (2) denoe he o Theorem 2 If ab ;ReaRebRe; hen 2R a b; ; ; 2Ra b; ; ; d (2) 2Rab ; ; ; d ; ; ; ; ; ; a b R ab R ab a b In ariular (22) b! ; ; ; ; ab ab a b; ; Proof d 2Rab ; ; ; d a a b b! a b b! a b b! R ab ; ; ; R ab ; ; ; (23) Coyrigh 23 SiRe

3 S B RAO ET A 337 whih i he (2) Now R a b; ; ; R a b; ; ; a b b! a b b! b a b b b a b a b b! b! a b b! a b b a a b! a b b! Thi i he roof of (22) or and ubiuing in above reul hi will immediaely lead o ariul ar ae (23) Theorem 2) If 2) If ab ; a b Re Re Re Re and hen u u R a b ; ; ; u du Rab ; ; ; (22) ab ; ReaRebReRe and hen R a b; ; ; R a b; ; ; d 3) In ariular Proof ) u ab ; Rea Reb Re and hen R a b; ; ; d R a b; ; ; ab ; ; d ab ; ; u u R a b; ; ; u u u du b! a b (222) (223) ( 224) a b u u d u b! a b b! R a b ; ; ; whih onlude he roof of (22) du Coyrigh 23 SiRe

4 338 S B RAO ET A 2) Therefore 2R ab ; ; ; d u u 2R ab ; ; ; u b R ab ; ; ; d alying he ranformaion formula u a b a b a b d u u u u du! d b! u u u b! 2Ra b; ; ; Whih i he roof of (222) 3) R a b; ; ; d R a b; ; ; a b a b 2Rab ; ; ; d d b! b! a b b! Thi lead he roof of (223) On uing in he above ereion immediaely lead o (224) Theorem 23 If ab ;Re a Re b Re Proof hen a b b R a b; ; ; d m 2Rab ; ; ; 2Rab ; m; ; d m m d a b 2 m m d 2Rab ; ; ; d d b! m m! m a b m b m! m m m 2R ab ; m; ; d (23) Coyrigh 23 SiRe

5 S B RAO ET A 339 Thi eablihe (23) 3 Rereenaion of Wrigh Tye R ab Hyergeomeri union in Term of he unio n Uing he definiion a b 2R a b; ; ; b we have R a b; ; ; a b a ; b ; ; ;! and aing bi a i! j! j b b b a ; ; where b ; i a -ule ; i a -ule b b b ; (3) Convergene rieria for generalied hyerfeomeri funion 2 ; 2 ; :! ) If he funion onverge for all finie 2) I f he funion onverge for and diverge for 3) If he funion i divergen for 4) If he funion i aboluely on- on he irle if vergen Re j i j i 4 Mellin-Barne Inegral Rereenaion of R ab Theorem 4 e ; ab ;Re a b Re Re Then 2R ab ; ; ; Barne inegral 2R ab ; ; ; i rereened by he Mellin- a b b a 2πi d (4) where arg π ; he onour of inegraion beginning a i and ending a i and inended o earae he ole of he inegrand a o he lef and all he ole a na n a well a n b n o he righ Proof We hall ue he um of reidue a he ole o obain he inegral of (4) ; a b b! b a R a b a b b a! b a (42) Now b ba π lim a b a 2πi a b re b a d a b a in π b b a (42) and (43) omlee he roof of (4) 5 Inegral Tranform of R ab (43) In hi eion we diued ome ueful inegral ranform lie Euler ranform alae ranfor m Mellin ranform and Whiaer ranform Theorem 5 (Euler (Bea) ranform) Coyrigh 23 SiRe

6 34 S B RAO ET A ; ; ; d 2Rab a b a b ; ; 32 ab a b where ;Re Re Re Re Re Re Proof R ab ; ; ; d (5) Re d b! a b a b d b! ab a b ; 32 ab ; a b! Thi i he roof of (5) Remar: Puing in (5) we ge ab ; ; d (52) a b ; ab ; 32 Taing and ubiuing in lae of he noaion ; (5) re due o R ab ; ; ; d R a b; ; (53) Alo onidering and in (5) wih relaemen of by a 2 R we ge R a b; ; ; d R a b; ; Theorem 52 (alae ranform) e R ab ; ; ; d a b ; 3 ab ; (54) (52) ab a b where ;Re Re Re Re Re Re Re and Proof ab e R ab ; ; ; d a b e d b! a b d e!! a b a b a b 3 a b ; Thi i he roof of (52) Theorem 53 (Mellin ranform) R a b; ; ; d a b ab w here ab ;Re a Re b Re Re Re Proof Puing 2R ab ; ; ; ; in (4) we ge ab b a 2πi a b 2πi b a d 2πi f where a b ab f (53) d d (532) Uing (9) () and (532) immediaely lead o (53) Coyrigh 23 SiRe

7 S B RAO ET A 34 Theorem 54 (Whiaer ranform) 2 e W R ab ; ; ; d a b ; ab where ab ;Re a Re Re Re Re b Re (54) Proof To obain Whiaer ranform we ue he following inegral: 2 v e W d v v 2 2 v where Rev 2 Subiuing v on he HS of (54) i redue o v v v 2 e W v R ab ; ; ; dv a b a b v 2 e v w vdv! a b a b 2 2! a b ; ab ; Thi omlee he roof of (54) 6 Relaionhi wih Some Known Seial union (o H-union Wrigh Hyergeomeri union) 6 Relaionhi wih o H-union Uing (4) we ge 2 H22 b a 2R a b; ; ; a b a b 2πi b a d 62 Relaionhi wih Wrigh Hyergeomeri union The Generalied Hyergeomeri union 2R ab ; ; ; a in (3) i a b b! 2R 2 R a b; ; ; ; rom () and (62) yield ; ; ; a b ; R R a b (62) a b ; (622) 7 Anowledgemen The auhor are hanful o he reviewer for heir valuable uggeion o imrove he ualiy of aer REERENCES [] N Virheno S Kalla and A Al-Zamel Some Reul on a Generalied Hyergeomeri union Inegral Tranform and Seial union Vol 2 No doi:8/ [2] E D Rainville Seial union The Mamillan Comany New Yor 96 [3] A Erdelyi e al Higher Tranendenal union MGaw-Hill New Yor [4] E M Wrigh On he Coeffiien of Power Serie Having Eonenial Singulariie Journal ondon Mahemaial Soiey Vol -8 No doi:2/jlm/-87 [5] M Doeno On Some Aliaion of Wrigh Hyergeomeri union Come Rendu de l Aadémie Bulgare de Siene Vol [6] V Maloviho On a Generalied Hyergeomeri union and Some Inegral Oeraor Mahemaial Phyi Vol [7] Galue A Al-Zamel and S Kalla urher Reul on Generalied Hyergeomeri union Alied Ma- hemai and Comuaion Vol 36 No doi:6/s96-33(2) 4- [8] I N Sneddon The Ue of Inegral Tranform Taa MGraw-Hill Publiaion Co d New Delhi 979 [9] H M Srivaava and H Manoha A Treaie on Coyrigh 23 SiRe

8 342 S B RAO ET A Generaing union John Wiley and Son/Elli Horwood New Yor/Chiheer 984 [] A M Mahai R K Saena and H J Haubold The H-union Sringer Berlin 2 doi:7/ Coyrigh 23 SiRe