Certain fractional derivative formulae involving the product of a general class of polynomials and the multivariable H-function

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1 Proc Indan Acad Sc Math Sc Vol 2 No 4 November 2002 pp Prnted n Inda Certan fractonal dervatve formulae nvolvng the product of a general cla of polynomal and the multvarable -functon R C SONI and DEEPIKA SING Department of Mathematc MN Inttute of Technology Japur Inda MS receved 5 Augut 2000 reved 4 June 2002 Abtract In the preent paper we obtan three unfed fractonal dervatve formulae FDF The frt nvolve the product of a general cla of polynomal and the multvarable -functon The econd nvolve the product of a general cla of polynomal and two multvarable -functon and ha been obtaned wth the help of the generalzed Lebnz rule for fractonal dervatve The lat FDF alo nvolve the product of a general cla of polynomal and the multvarable -functon but t obtaned by the applcaton of the frt FDF twce and t nvolve two ndependent varable ntead of one The polynomal and the functon nvolved n all our fractonal dervatve formulae a well a ther argument whch are of the type ρ = t σ + α are qute general n nature Thee formulae bede beng of very general character have been put n a compact form avodng the occurrence of nfnte ere and thu makng them ueful n applcaton Our fndng provde nteretng unfcaton and etenon of a number of new and known reult For the ake of llutraton we gve here eact reference to the reult n eence of fve reearch paper that follow a partcular cae of our fndng In the end we record a new fractonal dervatve formula nvolvng the product of the ermte polynomal the Laguerre polynomal and the product of r dfferent Whttaker functon a a mple pecal cae of our frt formula Keyword Remann Louvlle and Erdély Kober fractonal operator; fractonal dervatve formulae; general cla of polynomal; multvarable -functon; generalzed Lebnz rule Introducton We hall defne the fractonal ntegral and dervatve of a functon f0 pp ee alo 6 8 a follow: Let α β and γ be comple number The fractonal ntegral Reα > 0 and dervatve Reα < 0 of a functon fdefned on 0 gven by α β t α F α + β γ;α; t ftdt Ɣα 0 I αβγ Reα > 0 0 f= d q d q I α+qβ qγ q 0 freα 0 0 < Reα + q q = 2 3 where F the Gau hypergeometrc functon 55

2 552 R C Son and Deepka Sngh The operator I nclude both the Remann Louvlle and the Erdély Kober fractonal operator a follow: The Remann Louvlle operator I α αγ 0 f= t α ftdt Ɣα 0 R0 α f= Reα > 0 d q 2 d q Rα+q 0 freα 0 0 < Reα + q q = 2 3 The Erdély Kober operator E αγ 0 f=iαoγ 0 f= α γ Ɣα 0 t α t γ ftdtreα > 0 Alo Sn m occurrng n the equel denote the general cla of polynomal ntroduced by Srvatava p eq n/m Sn m = k=0 n mk A nk k n = 02 4 k! where m an arbtrary potve nteger and the coeffcent A nk n k 0 are arbtrary contant real or comple On utably pecalzng the coeffcent A nk S m n yeld a number of known polynomal a t pecal cae Thee nclude among other the ermte polynomal the Jacob polynomal the Laguerre polynomal the Beel polynomal the Gould opper polynomal the Brafman polynomal and everal other 6 pp 58 6 The -functon of r comple varable z z r wa ntroduced by Srvatava and Panda 5 We hall defne and repreent t n the followng form 4 p 25 eq C : where w = z z r = 0N:M N ; ;M r N r PQ:P Q ; ;P r Q r z a ; α αr P : c γ P ; ; z r b ; β βr Q : d Q δ ; ; = 2πω r L φ ξ = c r d r γ r P r δ r Q r L r φ ξ φ r ξ r ψξ ξ r z ξ zξ r r dξ dξ r M = Ɣ d Q =M + Ɣ d δ N ξ + δ ξ = Ɣ c + γ ξ P =N + Ɣ c γ ξ 3 5 { r} 6

3 Unfed fractonal dervatve formulae 553 ψξ ξ r = N= Ɣ a + r = α P=N+ Ɣ a r = α Q ξ = Ɣ b + r = β ξ ξ 7 The nature of contour L L r n 5 the varou pecal cae and other detal of the above functon can be found n the book referred to above It may be remarked here that all the Greek letter occurrng n the left-hand de of 5 are aumed to be potve real number for tandardzaton purpoe The defnton of th functon wll however be meanngful even f ome of thee quantte are zero Agan t aumed that the varou multvarable -functon occurrng n the paper alway atfy ther approprate condton of convergence 4 pp eq C5 and C6 2 Man reult 2 Fractonal dervatve formula I αβγ 0 { ρ z u = = α σ ασ n t /m t k t =0 e k ek t t + α σ t = S m n e λ t v + α z r u r = ρ β n /m k =0 n m k n t mt k t k!k t! = = A n k A t n t k t t η + α t +α v r t α η k + +η t k t α η k + +η t k t λ k + +λ t k t 0N++2:M N ; ;M r N r ;0; ;0 P ++2Q++2:P Q ; ;P r Q r 0; ;0 ; } z α v α v u z r α vr α vr α t α t ρ λ k λ t k t ;u u r t t β γ ρ λ k λ t k t ;u u r t t β ρ λ k λ t k t ;u u r t t α γ ρ λ k λ t k t ;u u r t t u r

4 554 R C Son and Deepka Sngh provded that + σ + η k + +η t k t; v vr +σ +η k + +η t k t ; v vr +σ +η k + +η t k t; v vr +σ +η k + +η t k t ; v vr a ;α αr b ;β βr : c γ P P ; ; : d Q δ ; ; Q c r γ r ; P r ; ; d r δ r ; Q r 0 ; ;0 8 Reα > 0; the quantte t t λ η η λ tη t ηt u v v u rv r vr are all potve ome of them may however decreae to zero provded that the reultng ntegral ha a meanng Reρ + r = u mn Re d M /δ + > 0 Alo the number occurrng below the lne at any place on the rght-hand de of 8 and throughout the paper ndcate the total number of zero/one/par covered by t Thu 0; ;0 would mean r zero/r one/r par and o on r r r 22 Fractonal dervatve formula 2 I αβγ 0 { ρ z u = t + α σ t = S m n e λ t v + α z r u r = = = = t η + α t +α v r z r+ u r+ r+ t v + α z r+τ u r+τ = α σ ασ ρ β l=0 n /m k =0 n t /m t k t =0 β l = } r+τ t v +α n m k n t mt k t k!k t!

5 Unfed fractonal dervatve formulae 555 A n k A t n t k t e k ek t t α η k + +η t k t α η k + +η t k t λ k + +λ t k t 0N+N +2+3:M N ; ;M r N r ;0; ;0;M r+ N r+ ; ;M r+τ N r+τ ;0; ;0 P +P +2+3Q+Q +2+3:P Q ; ;P r Q r 0; ;0 ; ;P r+ Q r+ ; ;P r+τ Q r+τ 0; ;0 ; z α v α v u z r α vr α vr α t α t z r+ α vr+ u r α vr+ z r+τ α vr+τ α t α t λ k λ t k t ;u u r t t u r+ α vr+τ τ + l γ λ k λ t k t ;u u r t t l λ k λ t k t ;u u r t t τ + α γ λ k λ t k t ;u u r t t +η k + +η t k t; v vr τ σ +η k + +η t k t ; v vr +η k + +η t k t; v vr 00 τ +2 +σ +η k + +η t k t ; v vr a ;α αr ρ; τ +2 N u r+ u r+τ t t r + τ + u r+τ τ + τ +2 τ +

6 556 R C Son and Deepka Sngh b ; β βr τ +2 Q β l ρ; β l γ; +σ ; α γ ρ; +σ ; +σ ; a + σ ; b u r+ u r+τ t t r + u r+ u r+τ t t r + v r+ r + v r+τ 2 u r+ u r+τ t t r + v r+ r + v r+τ r + ;0 0 r + r + ;0 0 r + v r+ vr+τ α r+ α r+τ v r+ vr+τ β r+ β r+τ 2 P Q a ; α αr : c τ +2 γ ; ; N+P P : c r c r+ d r+ γ r d δ γ r+ δ r+ ; ; ; ; P r Q ; ; P r+ ; ; Qr+ ; ; d r δ r Q c r+τ 0 ; ;0 ; ; r γ r+τ d r+τ δ r+τ Q 0; ;0 ; r+τ 9 ; ; ; P r+τ here z r+ z r+τ tand for the followng multvarable -functon of τ comple varable z r+ z r+τ 4 p 25 eq C:

7 Unfed fractonal dervatve formulae 557 z r+ z r+τ = 0N :M r+ N r+ ; ;M r+τ N r+τ P Q :P r+ Q r+ ; ;P r+τ Q r+τ z r+ z r+τ a ;αr+ b ;βr+ α r+τ β r+τ P : Q : c r+ γ r+ P c r+τ γ r+τ P r+τ d r+ δ r+ Q d r+τ ; ; r+ ; ; r+ δ r+τ Q r+τ The functon occurrng on the rght-hand de of 9 the -functon of r τ varable provded that Reα > 0; the quantte t t λ η η λ tη t ηt u v v u rv r vr u r+ v r+ v r+ u r+τv r+τ v r+τ are all potve ome of them may however decreae to zero provded that the reultng ntegral ha a meanng Reρ + r+τ = u mn Re d M /δ + > 0 23 Fractonal dervatve formula 3 { I αβγ 0 I α β γ 0y ρ y ρ e λ y ζ z u y u = = = t +α v r = t + α η t + α σ y t + β σ τ y t + β t = S m n t + α v y t + β w z r u r y u r } r w y t + β = α σ n /m ασ βσ βσ ρ β y ρ β n t /m t k t =0 e k ek t n m k n t mt k t k!k t! k =0 t α η k + +η t k t α η k + +η t k t λ k + +λ t k t y ζ k + +ζ t k t A n k A t n t k t β τ k + +τ t k t β τ k + +τ t k t 0

8 558 R C Son and Deepka Sngh 0N+2+4:M N ; ;M r N r ;0; ;0 P +2+4Q+2+4:P Q ; ;P r Q r 0; ;0 ; 2 z α v α v β w β w u y u z r α vr α vr α t α t β yt β y t ρ λ k λ t k t ;u u r β wr u r y u r β wr β γ ρ λ k λ t k t ;u u r β ρ λ k λ t k t ;u u r α γ ρ λ k λ t k t ;u u r +σ +η k + +η t k t; v vr +σ +η k + +η t k t ; v vr +σ +η k + +η t k t; v t t t t t t t t 2 vr 2 +σ +η k + +η t k t ; v vr 2 ρ ζ k ζ t k t ;u u r t 0 t 0 β γ ρ ζ k ζ t k t ;u u r t 0 t 0 β ρ ζ k ζ t k t ;u u r t 0 t 0 α γ ρ ζ k ζ t k t ;u u r t 0 t 0

9 provded that Unfed fractonal dervatve formulae σ + τ k + +τ t k t; w wr +σ +τ k + +τ k t;w wr +σ +τ k + +τ t k t; w wr +σ +τ k + +τ k t;w wr a ;α αr b ;β βr 2 c r P γ r 2 d r δ r : c γ ; ; P ; ; ; P r Q : d δ ; ; Q 0 ; ;0 ; Q r 2 Reα > 0; Reα > 0; the quantte t t t t λ η η λ t η t ηt ζ τ τ ζ tτ t τt u v v u w w u r v r vr u r wr wr are all potve ome of them may however decreae to zero provded that the reultng ntegral ha a meanng Reρ + r = u mn Re d M ı /δ + > 0 and Reρ + r = u mn M ı Re /δ + > 0 d Proof of 8 To prove the fractonal dervatve formula FDF we frt epre the product of a general cla of polynomal occurrng on t left-hand de n the ere form gven by 4 replace the multvarable -functon occurrng theren by t well-known Melln Barne contour ntegral gven by 5 nterchange the order of ummaton ξ ξ r - ntegral and takng the fractonal dervatve operator nde whch permble under the condton tated wth 8 and make a lttle mplfcaton Net we epre the term t σ +η + α k + +η t k t v ξ v r ξr t σ +η +α k + +η t k t v ξ v r ξ r o obtaned n term of Melln Barne contour ntegral 4 p 8 eq 264; p 0 eq 2 Now nterchangng the order of ξ r+ ξ r+ and ξ ξ r -ntegral whch alo permble under the condton tated wth 8 and evaluatng the -ntegral thu obtaned by ung the known formula 9 p 6 Lemma I αβγ 0 { λ } = Ɣ + λɣ β + γ + λ Ɣ β + λɣ + α + γ + λ λ β Reλ > ma0 Reβ γ 2 and renterpretng the multvarable Melln Barne contour ntegral o obtaned n term of the -functon of r + varable we ealy arrve at the dered formula 8 after a lttle mplfcaton

10 560 R C Son and Deepka Sngh Proof of 9 To prove FDF 2 we take and f= ρ t σ +α = z r+ u r+ r+ t v +α = g = t + α σ = r+τ t v +α z u t = S m n e λ = t v + α z r u r = z r+τ u r+τ t η + α = r t v +α n the left-hand de of 9; and apply the followng generalzed Lebnz rule for the fractonal ntegral I αβγ 0 {fg} = l=0 β l I αβ lγ 0 {f}i αlγ 0 {g} 3 we ealy obtan FDF 2 after a lttle mplfcaton on makng ue of FDF and a known reult 4 p 9 eq 6 Proof of To prove FDF 3 we ue the formula FDF twce frt wth repect to the varable y and then wth repect to the varable ; here and y are ndependent varable 3 Specal cae and applcaton The fractonal dervatve formulae 2 and 3 etablhed here are unfed n nature and act a key formulae Thu the general cla of polynomal nvolved n FDF 2 and 3 reduce to a large pectrum of polynomal lted by Srvatava and Sngh 6 pp 58 6 and o from formulae 2 and 3 we can further obtan varou fractonal dervatve formulae nvolvng a number of mpler polynomal Agan the multvarable -functon occurrng n thee formulae can be utably pecalzed to a remarkably wde varety of ueful functon or product of everal uch functon whch are epreble n term of EFG and -functon of one two or more varable For eample f N = P = Q = 0 or N = P = Q = 0 the multvarable -functon occurrng n the left-hand de of thee formulae would reduce mmedately to the product of r or τ dfferent -functon of Fo Thu the table ltng varou pecal cae of the -functon 5 pp can be ued to derve from thee fractonal dervatve formulae a number of other FDF nvolvng any of thee mpler pecal functon On reducng the operator defned by to the Remann Louvlle operator gven by 2 we arrve at three fractonal dervatve formulae nvolvng thee operator but we do not record them here eplctly Agan our FDF 2 and 3 wll alo gve re n eence to a

11 Unfed fractonal dervatve formulae 56 number of other FDF lyng cattered n the lterature ee 2 pp eq pp eq pp 7 72 eq 2 and 2 p 7 eq 3 on makng utable ubttuton Alo f we take σ = 0 = ν 8 the polynomal S m 0 Sm t = = νr = and n = 0 = t n 0 wll reduce to A 00 At 00 repectvely whch can be taken to be unty wthout lo of generalty we arrve at the formula gven by 0 p 532 eq 4 If n FDF we take t = 2 and reduce the polynomal S m n to the ermte polynomal 6 p 58 eq 4 the polynomal S m 2 n 2 to the Laguerre polynomal 6 p 59 eq 8 the multvarable -functon to the product of r dfferent Whttaker functon 4 p 8 eq 267 we arrve at the followng new and nteretng pecal cae of the FDF after a lttle mplfcaton I αβγ 0 = k r b l + n 2 ρ+ l= = r I= t σ + α n 2 L θ n 2 } ep z l 2 W µl ν l z l rl= z l b l α σ ασ ρ β n /2 Ɣ σ Ɣ σ n2 + θ n 2 k +k 2 θ + k2 n 2 k =0 k 2 =0 n 2k n 2 k2 k!k 2! z z r ; ; 02:20; ;20;; ; 2; ;2 22: r ; α t α t ρ k k 2 ; t t β γ ρ k k 2 ; β ρ k k 2 ; t t r t t : α γ ρ k k 2 ; t t : r b µ +; ;b r µ r +;+σ ; ;+σ b ±ν + 2 ; ; b r ±ν r + 2 0; ;0 ; The condton of valdty of 4 can be ealy obtaned from thoe of 8 Several other nteretng and ueful pecal cae of our man fractonal dervatve formulae 2 and 3 nvolvng the product of a large varety of polynomal whch are pecal cae of S m n S m t n t and numerou mple pecal functon nvolvng one or more 4

12 562 R C Son and Deepka Sngh varable whch are partcular cae of the multvarable -functon can alo be obtaned but we do not record them here for lack of pace Acknowledgement The author are thankful to the referee for h ueful uggeton Reference Fo C The G and -functon a ymmetrcal Fourer kernel Tran Amer Math Soc Gupta K C and Agrawal S M Fractonal ntegral formulae nvolvng a general cla of polynomal and the multvarable -functon Proc Indan Acad Sc Math Sc Gupta K C Agrawal S M and Son R C Fractonal ntegral formulae nvolvng the multvarable -functon and a general cla of polynomal Indan J Pure Appl Math Gupta K C and Son R C A tudy of -functon of one and everal varable J Raathan Acad Phy Sc Matha A M and Saena R K The -functon wth applcaton n tattc and other dcplne New Delh: Wley Eatern Lmted Mller K S and Ro B An ntroducton to the fractonal calculu and fractonal dfferental equaton New York: John Wley and Son Oldham K B and Spaner J The fractonal calculu New York: Academc Pre Sago M A remark on ntegral operator nvolvng the Gau hypergeometrc functon Math Rep Kyuhu Unv Sago M and Rana R K Fractonal calculu operator aocated wth a general cla of polynomal Fukuoka Unv Sc Report Sago M and Rana R K Fractonal calculu operator aocated wth the -functon of everal varable n: Analy Geometry and Group: A Remann Legacy Volume ed M Srvatava and Th M Raa Palm arbor Florda USA adronc Pre ISBN Srvatava M A contour ntegral nvolvng Fo -functon Indan J Math Srvatava M Chandel R C Sngh and Vhwakarma P K Fractonal dervatve of certan generalzed hypergeometrc functon of everal varable J Math Anal Appl Srvatava M and Goyal S P Fractonal dervatve of the -functon of everal varable J Math Anal Appl Srvatava M Gupta K C and Goyal S P The -functon of one and two varable wth applcaton New Delh: South Aan Publher Srvatava M and Panda R Some blateral generatng functon for a cla of generalzed hypergeometrc polynomal J Rene Angew Math 283/ Srvatava M and Sngh N P The ntegraton of certan product of the multvarable -functon wth a general cla of polynomal Rend Crc Mat Palermo