International Journal of Pure and Applied Sciences and Technology
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1 It. J. Pure Appl. Sci. Techol. 6(2 (2 pp Iteratioal Joural of Pure ad Applied Sciece ad Techology ISSN Available olie at Reearch Paper Geeralized Fractioal Differetiatio of the -Fuctio Ivolvig Geeral Cla of Polyoial D. Kuar * ad J. Daiya Departet of Matheatic & Statitic Jai Narai Vya Uiverity Jodhpur 42 5 Idia * Correpodig author e-ail: (dieh_dio@yahoo.co (Received: 25-2-; Accepted: 4-4- Abtract: The obect of thi paper i to etablih two theore for the geeralized fractioal differetiatio of the product of a geeral cla of polyoial ad -fuctio. The coidered geeralized fractioal differetiatio operator cotai the Appell fuctio F [2 p. 224 Eq. 5.7.(8] a a erel ad are itroduced recetly by Saigo ad Maeda [4 p. 9]. It ha bee how that the geeralized fractioal derivative of the product of a geeral cla of polyoial ad -fuctio are trafored ito other -fuctio but of greater order. O accout of the geeral ature of the Saigo-Maeda operator -fuctio ad a geeral cla of polyoial a large uber of ew ad ow reult follow a pecial cae of our ai fidig. Soe iteretig pecial cae ivolvig variou pecial fuctio otably geeralized Wright hypergeoetric fuctio geeralized Wright-Beel fuctio the polylogarith geeralized Riea Zeta fuctio ad Whittaer fuctio are alo preeted. Keyword: Geeralized fractioal calculu operator -fuctio Geeralized Wright hypergeoetric fuctio Geeralized Wright-Beel fuctio the polylogarith fuctio Geeralized Riea Zeta fuctio ad Whittaer fuctio.. Itroductio ad Preliiarie A igificatly large uber of wor o the ubect of fractioal calculu give iteretig accout of the theory ad applicatio of fractioal calculu operator i ay differet area of atheatical aalyi. A lot of reearch wor ha bee recetly coe up o the tudy ad developet of a fuctio that i ore geeral tha Fo -fuctio ow a -fuctio.
2 It. J. Pure Appl. Sci. Techol. 6( (2 pp The -fuctio wa itroduced by Iayat-uai [ 4] it i defied ad repreeted i the followig aer [] M N M N ( a ; A ( a N N P ξ P Q [ z] P Q z ( z d ( ( ( z b b ; B M θ ξ ξ M Q 2π i (. L where i ad θ ( { } ( b ξ ( a ξ M N A Γ Γ Q B P { ( } Γ b ξ Γ( a ξ M N. (.2 ere a ( P ad b ( Q are cople paraeter ( P ( Q (ot all zero iultaeouly ad the epoet a ( N b ( M Q ad ca tae o iteger value reader ca alo refer the paper [ 4] for ore detail. The followig ufficiet coditio for the abolute covergece of the defiig itegral for the -fuctio give by Buha ad Srivatava [] M N Q P (. Ω A B > M N ad arg z < Ω π. 2 Alo fro Iayat-uai [] it follow that ζ ( M N * P Q z o z for all z b where ζ * i Re (.4 M a z o z for large z where ξ* a Re A. (.5 N M N ξ* ad P Q ( Whe the epoet A B i ad the -fuctio reduce to the failiar Fo - fuctio [8]. Alo S Srivatava [2]: occurrig i the equel deote the geeral cla of polyoial itroduced by ( [ ] 2 (.6 S A
3 It. J. Pure Appl. Sci. Techol. 6( (2 pp where i a arbitrary poitive iteger ad the coefficiet ( real or cople. O uitably pecialize the coefficiet polyoial a it pecial cae (ee Srivatava ad Sigh [2] pp A are arbitrary cotat A S yield a uber of ow Recetly geeralized fractioal calculu of the - fuctio give by Saigo ad Kilba [] Ra ad Kuar [] have obtaied the iage of the product of two -fuctio i Saigo-Maeda operator; Saea Ra ad Kuar [6] have obtaied the geeralized fractioal itegral forulae of the product of Beel fuctio of the firt id ivolvig Saigo-Maeda fractioal itegral operator. Siilarly geeralized fractioal calculu forulae of the Aleph-fuctio aociated with the Appell fuctio F i give by Saea et al. [7 8] ad Ra & Kuar []. 2. Geeralized Fractioal Differetiatio Operator The fractioal calculu operator ivolvig variou pecial fuctio have bee foud igificat iportace ad applicatio i variou ub-field of applicatio atheatical aalyi. Sice lat five decade a uber of worer lie Kiryaova [7] Srivatava et al. [2 22] Saea et al. [6-8] Saigo [2] Kilba [5] Sao Kilba ad Marichev [5] Miller ad Ro [9] ad Ra ad Kuar [ ] etc. have tudied i depth the propertie applicatio ad differet eteio of variou hypergeoetric operator of fractioal itegratio. Let γ C γ > ad R the the geeralized fractioal differetiatio operator [4] ivolvig Appell fuctio F a a erel are defied by the followig equatio: ( D γ ( ( f I γ f ( d d (2. ( I γ f ( γ ( γ ( Re( > ; Re (2.2 d γ ( ( γ t t t F ; f ( t dt Γ( γ d t (2. ad ( D γ f ( ( I γ f ( d d (2.4 ( Re( > ; Re ( I γ f ( γ ( γ (2.5 d γ ( ( γ t t t F ; f ( t dt Γ( γ d t (2.6 γ γ where I ad I are Saigo-Maeda fractioal itegral operator ad Appellhypergeoetric fuctio of two variable i defied a
4 It. J. Pure Appl. Sci. Techol. 6( (2 pp ( ( ( ( ( γ z ξ F ( ; γ ; z ξ ( z < ξ < ; (2.7 where ( z ad ( z are the Pochhaer ybol defied by z C N { 2 } by ( z ( z z ( z ( z. The erie i (2.7 i abolutely coverget for ( z < ξ < ad ( z ξ where ( z ξ. Thee operator reduce to Saigo derivative operator [4 9] a γ γ γ γ ( D f ( ( D f ( ( ( γ ad N N { } Re > ; (2.8 γ γ γ γ ( D f ( ( D f ( ( ( γ Re >. (2.9 Further [4 p. 94 Eq. (4.8 ad (4.9] we alo have ρ ρ γ ρ I Γ ρ γ ρ γ ρ γ ρ ρ γ where Re( γ > Re( ρ > a Re( γ Re( γ ρ γ ρ ρ I Γ ρ γ ρ ρ (2. ad γ ρ ρ γ where Re( γ > Re( ρ < i Re( Re ( γ Re( γ. (2. ere we have ued the ybol Γ repreetig the fractio of ay Gaa fuctio.. Mai Reult ( Theore : Let γ µ z C Re( γ > ad { 2 } cotat M N P Q N a ( P b ( Q z ad the epoet A ( N B ( M Q N a [ τ ζ *] < Re( µ i Re ( Re( γ followig relatio: >. Further let the are cople arg < π Ω 2 Ω > be give ad atify the coditio. The we have the ( a ; A ( a ( b ( b ; B M N N γ N P D µ t S c t P Q z t M M Q (
5 It. J. Pure Appl. Sci. Techol. 6( (2 pp ( ( [ ] [ ] µ γ ( ( A A c c M N P Q ( µ ( µ γ ( b ( b M B M Q ( µ γ ; ; z ; ; ( µ ;( ; ( a A a N N P ( µ γ ; ( µ ;. (. Proof: I order to prove (. we firt epre the product of a geeral cla of polyoial occurrig o it left-had ide i the erie fro give by (.6 replacig the -fuctio i ter of Melli- Bare cotour itegral with the help of equatio (. iterchage the order of uatio we obtai the followig for (ay I : [ ] ( ( ( I A A c c 2π i L µ ξ ξ γ θ ( ξ z D t ( dξ ( ( ( A A c c d d 2π i L µ γ ξ Q ( z { } ( b ξ ( a ξ M N A Γ Γ B P { ( } Γ b ξ Γ ( a ξ M N Γ ( µ ξ Γ ( µ ξ γ ( µ ξ Γ ( µ ξ γ ( µ ξ γ ( µ ξ Γ Γ Γ ( ( µ γ ( A ( A c c d d M N P Q ( µ ;( µ γ ; ( b ( b M B M Q ( µ γ z ; ; dξ
6 It. J. Pure Appl. Sci. Techol. 6( (2 pp ere ( γ ( µ ;( ; ( a A a N N P ( µ γ ; ( µ ;. Re ad by uig Γ ( ( d where i the above d Γ epreio we obtai the right-had ide of (.. Thi i coplete proof of Theore. I view of the relatio (2.8 the we arrive at the followig corollary cocerig Saigo fractioal derivative operator [2]. ( Corollary.: Let γ µ z CRe( > ad { 2 } coditio relatio: > ad atify the b Re ( µ i Re > a M Re( γ. The we have the followig ( a ; A ( a ( b ( b ; B M N N γ N P D µ t S c t P Q z t M M Q [ ] ( ( µ ( ( A A c c M N 2 P 2 Q 2 ( µ ( µ γ ( b ( b ; B ( µ ; M M Q ; ; z ( a ; A ( a N N P ( µ γ ; ( (.2 where the coditio of eitece of the above corollary follow eaily with the help of (.. Rear : We ca alo obtai reult cocerig Riea-Liouville ad Erdélyi-Kober fractioal derivative operator by puttig ad repectively i Corollary.. ( Theore 2: Let γ µ z C Re( γ > ad { 2 } M N P Q N a ( P b ( Q are cople cotat arg z < π Ω 2 atify the coditio >. Further let the Ω > ad the epoet ( ( A N B M Q N be give ad
7 It. J. Pure Appl. Sci. Techol. 6( (2 pp ( ( ( ( i τ ξ * > Re µ a Re γ Re Re γ where ( γ Re. The we have the followig relatio: ( a ; A ( a ( b ( b ; B M N N γ N P D µ t S ct P Q zt M M Q [ ] [ ] ( ( ( γ µ γ ( Re ( A ( A c c ( a ; A ( a ( µ ( γ µ ( γ µ z M N N N P P Q ( γ µ ( µ ( µ ( ( ; b b M B M Q (. (. Proof: I order to prove (. we firt epre the product of a geeral cla of polyoial occurrig o it left-had ide i the erie fro give by (.6 replacig the -fuctio i ter of Melli- Bare cotour itegral with the help of equatio (. iterchage the order of uatio we obtai the followig for (ay I : [ ] ( ( ( I A A c c 2π i L µ ξ ξ γ θ ( ξ z D t ( dξ ( ( ( A A c c d d 2π i L µ γ ξ Q ( z { } ( b ξ ( a ξ M N A Γ Γ B P { ( } Γ b ξ Γ( a ξ M N
8 It. J. Pure Appl. Sci. Techol. 6( (2 pp ( γ µ ξ ( γ µ ξ ( µ ξ Γ ( µ ξ Γ ( γ µ ξ Γ ( µ ξ Γ Γ Γ ere ( γ Re ad by uig Γ ( ( d where i the above d Γ epreio we obtai the right-had ide of (.. Thi i coplete proof of Theore 2. I view of the relatio (2.9 the we arrive at the followig corollary cocerig Saigo fractioal derivative operator. ( Corollary 2.: Let γ µ z CRe( > ad { 2 } coditio ( ( ( > ad atify the i τ ξ * > Re µ a Re γ Re.The we have the followig relatio: ( a ; A ( a ( b ( b ; B M N N γ N P D µ t S c t P Q z t M M Q [ ] [ ] ( ( ( µ ( Re ( A ( A c c z ( a ; A ( a ( µ ( γ µ ( µ ( γ µ ( b ( b ; B M 2 N N N P P 2 Q 2 ( M M Q Where the coditio of eitece of the above corollary follow eaily fro Theore 2. Rear 2: We ca alo obtai reult cocerig Riea-Liouville ad Erdélyi-Kober fractioal derivative operator by puttig ad repectively i Corollary Special Cae ad Applicatio (i If we tae M N P ad Q Q i Theore the -fuctio occurrig therei brea up ito the geeralized Wright hypergeoetric fuctio (. the followig for after a little iplificatio: P Q (.4 ψ give by []. The Theore tae dξ ( a ; A ( b ; B γ P D µ t S c t Pψ Q z t Q (
9 It. J. Pure Appl. Sci. Techol. 6( (2 pp [ ] ( ( µ γ ( ( A A c c P P Q 4 ( µ ( µ γ ( b ; B ( µ γ ; Q ; ; z ( µ ;( ; a A P ( µ γ ; ( µ ;. (4. The coditio of validity of the above reult eaily follow fro (.. Rear : If we et S ad A B ad ue the relatio (2.8 alo ae uitable adutet i the paraeter i (4. we arrive at the ow reult give by Kilba [5]. (ii If we tae M N P ad Q 2 i (. ad the -fuctio occurrig therei brea B up ito the geeralized Wright-Beel fuctio J b ( z B D t S ct J γ µ b zt [ ] ( ( ( (. The Theore tae the followig for: µ γ ( ( A A c c 5 ( µ ; ( µ γ ; ( ( b B ( µ γ ( µ γ z ; ; ; ( µ ; ( µ ;. (4.2 The coditio of validity of the above reult ca be eaily followed directly fro a give with (.. (iii If we tae M ad N P Q 2 i (. the -fuctio reduce ito the Poly- v logarith [8 p. 27 Eq. (A.65] of cople order v deoted by L ( z. The the equatio (. tae the followig for after iplificatio:
10 It. J. Pure Appl. Sci. Techol. 6( (2 pp γ µ v D t S c t L zt [ ] ( ( ( ( µ γ ( ( A A c c ( µ ( µ γ ( µ ; ( ( ; v ( µ γ ; ( µ γ ; ; ; z 5 55 ( ( ; v ( µ ;. (4. The coditio of validity of the above reult ca be eaily followed directly fro Theore. (iv If we reduce the -fuctio occurrig i the left-had ide of (. to geeralized Riea Zeta fuctio [2 p. 27 Eq. (] give by ( ( η; l ( ( η ; l z φ ( z l η z ( η r r 2 l 22 (4.4 r the we arrive at the followig reult: D γ µ t S c t φ zt l [ ] ( ( ( η ( µ γ ( ( A A c c ( µ ( µ γ ( µ ; ( ( η ; l ( µ γ ; ( µ γ ; ; ; z 5 55 ( ; ( η; l ( µ ;. (4.5 The coditio of validity of the above reult ca be eaily followed directly fro a give with (..
11 It. J. Pure Appl. Sci. Techol. 6( (2 pp (v If we et product of polyoial [2] by taig A B the we obtai the followig reult: zt γ µ 2 D t e Wa b zt S to uity ad reduce the -fuctio to Whittaer fuctio ( ( ( µ ; ( µ γ ; ( 2± b ( µ γ ( µ γ µ γ 2 45 z ; ; ( µ ; ( a ( µ ;. (4.6 The coditio of validity of the above reult ca be eaily followed directly fro Theore. (vi If we et product of polyoial S to uity ad reduce the -fuctio to Fo -fuctio the we ca eaily obtai the ow reult give by Saea ad Saigo [9]. 5. Cocluio I the preet paper we have give the two theore of geeralized fractioal derivative operator give by Saigo-Maeda. The theore have bee developed i ter of the product of -fuctio ad a geeral cla of polyoial i a copact ad elegat for with the help of Saigo-Maeda power fuctio forulae. Mot of the give reult have bee put i a copact for avoidig the occurrece of ifiite erie ad thu aig the ueful i applicatio. Referece [] R.G. Bucha ad.m. Srivatava The -fuctio aociated with a certai cla of feya itegral J. Phy. A. Math. Ge. 2( [2] A. Erdélyi W. Magu F. Oberhettiger ad F.G. Tricoi igher Tracedetal Fuctio (Vol. I McGraw-ill New Yor- Toroto-Lodo Reprited: Krieger Melboure Florida 95. [] A.A. Iayat-uai New propertie of hypergeoetric erie derivable fro feya itegral: I Traforatio ad reeducatio forulae J. Phy. A. Math. Ge. 2( [4] A.A. Iayat-uai New propertie of hypergeoetric erie derivable fro feya itegral: II A geeralizatio of the -fuctio J. Phy. A. Math. Ge. 2( [5] A.A. Kilba Fractioal calculu of the geeralized wright fuctio Fract. Calc. Appl. Aal. 8(2 ( [6] A.A. Kilba ad M. Saigo -Trafor Theory ad Applicatio Chapa ad all/crc Pre Boca Rato FL. 24.
12 It. J. Pure Appl. Sci. Techol. 6( (2 pp [7] V. Kiryaova A brief tory about the operator of the geeralized fractioal calculu Fract. Calc. Appl. Aal. (2 ( [8] A.M. Mathai R.K. Saea ad.j. aubold The -Fuctio: Theory ad Applicatio Spriger New Yor 2. [9] K.S. Miller ad B. Ro A Itroductio to Fractioal Calculu ad Fractioal Differetial Equatio Joh Wiley ad So Ic. New Yor 99. [] J. Ra ad D. Kuar Geeralized fractioal itegratio ivolvig Appell hypergeoetric of the product of two -fuctio Viaaa Prihad Auadha Patria 54( (2-4. [] J. Ra ad D. Kuar Geeralized fractioal itegratio of the -fuctio J. Raatha Acad. Phy. Sci. (4 ( [2] M. Saigo A rear o itegral operator ivolvig the gau hypergeoetric fuctio Math. Rep. College Geeral Ed. Kyuhu Uiv. ( [] M. Saigo ad A.A. Kilba Geeralized fractioal calculu of the - fuctio Fuuoa Uiv. Sci. Rep. 29( [4] M. Saigo ad N. Maeda More Geeralizatio of Fractioal Calculu: Trafor Method ad Special Fuctio Vara Bulgaria 996. [5] S.G. Sao A.A. Kilba ad O.I. Marichev Fractioal Itegral ad Derivative Theory ad Applicatio Gordo ad Breach Sci. Publ. New Yor 99. [6] R.K. Saea J. Ra ad D. Kuar Geeralized fractioal itegratio of the product of Beel fuctio of the firt id Proceedig of the 9th Aual Coferece SSFA 9( [7] R.K. Saea J. Ra ad D. Kuar Geeralized fractioal differetiatio for Saigo operator ivolvig Aleph-fuctio J. Idia Acad. Math. 4( ( [8] R.K. Saea J. Ra ad D. Kuar Geeralized fractioal differetiatio of the Alephfuctio aociated with the Appell fuctio F Acta Ciecia Idica 8(4 ( [9] R.K. Saea ad M. Saigo Geeralized fractioal calculu of the -fuctio aociated with the Appell fuctio F J. Fract. Calc. 9( [2].M. Srivatava A cotour itegral ivolvig fo -fuctio Idia J. Math. 4( [2].M. Srivatava K.C. Gupta ad S.P. Goyal The -Fuctio of Oe ad Two Variable with Applicatio South Aia Publiher New Delhi/Madra 982. [22].M. Srivatava R.K. Saea ad J. Ra Soe ultidieioal fractioal itegral operatio ivolvig a geeral cla of polyoial J. Math. Aal Appl. 9( [2].M. Srivatava ad N.P. Sigh The itegratio of certai product of the ultivariable - fuctio with a geeral cla of polyoial Red. Circ. Mat. Palero 2(
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