Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus
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1 athematics Letters 06; (6: doi: 068/ml06006 Relatios Amog Certai Geeralied Hper-Geometric uctios Suggested b N-fractioal Calculus aged Gumaa Bi-Saad Departmet of athematics Ade Uiversit Ade Kohrmasar Yeme address: mgbisaad@ahoocom To cite this article: aged Gumaa Bi-Saad Relatios Amog Certai Geeralied Hper-Geometric uctios Suggested b N-fractioal Calculus athematics Letters Vol No 6 06 pp 7-57 doi: 068/ml06006 Received: August 7 06; Accepted: November 9 06; Published: December 0 06 Abstract: The subect of fractioal calculus has gaied importace ad popularit durig the past three decades Based upo the N-fractioal calculus we itroduce a ew N-fractioal operators ivolvig hper-geometric fuctio B meas of these N- fractioal operators a umber of operatioal relatios amog the hper-geometric fuctios of two three four ad several variables are the foud Other closel-related results are also cosidered Kewords: N-fractioal Calculus Operators Hor s uctios Appell uctios Sara uctios Quadruple uctios Hper-Geometric of Several Variables Itroductio The subect of fractioal calculus is oe of the most itesivel developig areas of mathematical aalsis mail due to its fields of applicatio rage from biolog through phsics ad electrochemistr to ecoomics probabilit theor special fuctios ad statistics (see [6] ad [7] Ideed o behalf of the ature of their defiitios the fractioal derivatives ad itegrals provide a ecellet istrumet for the modelig of memor ad hereditar properties of various materials ad processes Half-order derivatives ad itegrals prove to be more useful for the formulatio of certai electrochemical problems tha the classical methods [9] Appell (see []; also [ p -] defied the four hper-geometric fuctios of two variables which he deoted b ad Other hpergeometric fuctios of two variables has bee defied b Hor [6] Seve of them he deoted b H H H H H H H (see eg [ p ] ad ad [ p 56-57] Appell s ad Hor s fuctios are all geeraliatio of Gaussia hpergeometric fuctio ( a ( b m [ a b; c; m m ] = ( m= 0 ( c m m! Γ ( a m where ( a m = Γ : Gamma fuctio Γa I 89 Lauricella [5] further geeralied the four Appell s fuctios ad to fuctios of - variables ( ( ( ( c ad A B D respectivel After a gap of log time Sara [8] iitiated a sstematic stud of te series of three variables with the otatios ad I 98 Eto [8] has K N P R S studied 0 triple hper-geometric series He deoted four of them b X X X X Srivastava ad Karlsso [] 6 8 ad 7 T provide a impressive tabulatio ad a wealth of iformatio o the costructio of the set of all 05 distict triple Gaussia hpergeometric series I their wor smbols ad refereces are give for those series that have bee itroduced previousl ad the deoted the ew series b the smbol (cf [ p 7]: ; ; ; ; or istace the series m p= 0 d has the defiitio ( a m ( b m ( c ( d p ( e p m m p ( f ( g p m!! p! E G
2 athematics Letters 06; (6: ad it would be preseted i the form a b ; a c ; d e e ; f ; g Sharma ad Parihar [0] itroduced 8 complete hpergeometric series of four variables It is 8 remarable that out of these 8 series the followig 9 series had alread bee appeared i the literature due to Eto (see [9] i the differet otatios: = K = K = K = K = K = K = K ( = K = K = K = K = K = K = K or the purpose of this wor we recall here some defiitios Defiitio (B K Nishimoto (see [6] ad [7] Let D C D = { D C = { C C = K = K 79 = K = K = K 6 be a curve alog the cut oiig two poits ad ilm( C be a curves alog the cut oiig two poits ad ilm( D be a domai surrouded b C D be a domai surrouded b C oreover let f = f( be aaltic (regular fuctio i D ad it has o brach poit iside C ad o C ad f ( ζ ( ζ v Γ ( v f = ( f ( = d v v ζ πi C ν Z lim ( f ( f = ν ν ( Z ( where π arg( ζ π for C - 0 arg( ζ π for v C ζ the ( f ν is the fractioal differ-itegratio of arbitrar order v (derivatives of order v for v>0 ad itegrals of order v for v<0 with respect to of the fuctio f if ( f ν < ν urthermore let N be Nishimoto s operator defied b [Refer to (] ν Γ ν dζ = πi C ( ν ν Z ( ζ N ( m ν with N = lim N m Z ν m Lemma We have -iπ Γ( Γ( ( = e < ( Γ( Γ( Operatioal represetatios ad relatios ivolvig oe ad more variables hpergeometric series have bee give cosiderable i the literature see for eample Che ad Srivastava [5] Goal Jai ad Gaur ([7] [8] Kalla [] Kalla ad Saea ([] ad [] Kat ad Koul [] Cha ad Srivastava [5] The preset sequel to these earlier papers is motivated largel b the aforemetioed wor of Bi-Saad ad aisoo [] i which a umber of operatioal relatios amog the hpergeometric fuctios of two ad three variables are foud The aim of this paper is to itroduce some N-fractioal operators ivolvig certai hpergeometric fuctios Based upo these operators we aim here to derive operatioal relatios amog the above said hpergeometric fuctios of two three four ad multiple variables The structure of this paper is the followig: I sectio we establish N- fractioal operators Sectio deals with same applicatios of these operators ad the presetatio of operatioal relatios betwee fuctios of two three ad four variables Sectio aims at itroducig multivariable geeraliatio of the N-fractioal operators i sectio ad establishig some operatioal relatios amog hpergeoemtric fuctios of several variables N-fractioal Calculus Operators B usig defiitios ( ad ( we itroduce three ids of N-fractioal operators i the followig defiitios: Defiitio Let be a N-fractioal operator defied b = N { ( ( = ( Γ( ζ ( ζ = d C πi ( ζ (5 ζ where C ad ( Defiitio Let operator defied b ; Z w be a N-fractioal ; w { ( { ( w w w ( w ( w ( ( = = ( ( w
3 9 aged Gumaa Bi-Saad: Relatios Amog Certai Geeralied Hper-Geometric uctios Suggested b N-fractioal Calculus Γ( Γ( ζ ( ζ ξ ( ξ dζ dξ = ( C C ( πi ζ ( ξ w where { C ad {( ( ; w w (6 Z Defiitio Let operator defied b = w be a N-fractioal ; { w ( w { ( w { f ( ( w { f ( w ( w ( = ( ( w ζ ζ ( ξ ( ξ Γ( Γ( ( ξ = { f dζ dξ C C ( i ( π (7 ζ ( ξ w where { C ad {( ( I view of lemma ad the biomial theorem the operators ( m m ( = m = 0 m! ca be writte i the forms: ad ; w ; w Z ad A [ ; ; ] ; w = (8 = B [ ; ; ] [ ; ; w] (9 w = ; ( ( m B w ( m m [ m ; ; ] (0 m= 0 m! m respectivel where throughout this wor ad ( ( A e iπ Γ = Γ( ( O replacig ad b - ad - respectivel relatio (8 reduces to a ow result due to Nishimoto [9 p 78 (] urther if = = relatio (0 ields w = B [ ; ; w] ; ( Whereas if = = relatio (0 ields w = B [ ; w] ; where (] ; is Appell s fuctio of two variables [ p Applicatios B choosig a suitable hper-geometric fuctio the operators (5 (6 ad (7 cab be applied to deduce relatioships ivolvig a fairl variet of hper-geometric fuctios of oe ad more variables i Relatios amog two ad three variables fuctios B maig use of the N-fractioal operator (5 we establish the followig relatioships betwee hpergeometric series of two ad three variables B = e w ( iπ ( ( ( Γ Γ Γ( Γ( { λ σ ; µ ; A [ λ λ σ ; µ µ ; ] = (5 { δ σ µ [ δ σ ; µ µ ; ] ; ; (6 G { λ σ ; µ η ; [ λ λ σ ; µ η ; ] (7 K ( { σ δ µ η [ σ δ ; µ η ; ] ; ; (8 A { λ σ δ ; µ ; [ λ δ σ µ µ ] (9 N ; ;
4 athematics Letters 06; (6: { σ ; ; δ µ = A X0 [ σ δ µ ] ( (0 ; ; { λ ; µ η ; A [ λ λ ; µ η ; ] E = ( { [ λ σ ; µ - ; ( ] 8c λ σ ; λ σ ; µ ; ; ( { H λ σ ; µ ; 0 d σ λ ; σ ; µ ; λ ; ( { H λ σ ; µ ; 7 b λ ; σ ; µ ; λ ; { H [ λ σ δ ] ; ; ( i σ λ ; σ δ ; ; λ ; (5 ; σ λ ; δ ε H c ; ; λ { [ λ σ δ ε ; ; ( ] (6 { [ σ δ ε ; µ ; ( ] H = A b σ ; ; δ ε µ ; ; (7 { H λ σ δ ; µ ; { H σ ; µ ; δ ; ; σ λ d λ; ; µ ( A X [ ] 6 σ ; µ ; { H λ ; µ ; (8 = (9 A X [ ] λ ; µ ; = (0 λ λ ; λ σ ; H 0d ; ; { [ λ σ ; ; ( ( ] ( { H ; ; = A X [ σ µ η ] σ µ η ( ( 8 ; ; { H λ ; µ η ; A X [ ] 7 λ ; µ η ; = ( λ λ ; λ σ ; H 9e ; µ ; { [ λ σ ; µ ; ( ] λ λ ; λ σ ; H 9d µ ; ; { [ λ σ ; µ ; ( ] ( (5 { H σ ; µ ; 5 ( b ; σ ; σ ; µ ; (6 { H λ ; µ ; ( 5 6c λ λ ; λ ; σ ; µ ; (7
5 5 aged Gumaa Bi-Saad: Relatios Amog Certai Geeralied Hper-Geometric uctios Suggested b N-fractioal Calculus { H [ λ σ ] 5 ; ; ( 9 λ λ ; λ ; ; µ ; (8 { H σ δ ; ( 6 ( 9 g ; ; δ σ σ ; ; (9 { H λ σ ; ( 6 6 c σ ; ; λ λ λ ; ; (0 { H λ σ ; 6 6 d σ ; ; λ λ λ ; ; σ H A ; ; σ δ = 7 ( 9b µ ; ; { σ δ ; µ ; ( { H [ λ σ δ ] 7 ; ; ( c σ δ ; ; λ λ λ ; ; ( PROO To establish relatio (5 we first write Appell s fuctio i it s series form (see eg [ p (]: ( λ ( ( m ; ; m m = m = 0 ( µ m m!! m!! [ λ µ ] Iterchage the order of summatio ad differ-itegratio which is permissible due to the absolute covergece we get { λ σ ; µ ; ( λ ( ( m m m = ( m = 0 ( µ m m!! m!! { Now maig use of (5 the idetit ( a m ( a ( a m = ad the defiitio of Sara s fuctio [8 p (5] with a little simplificatio the desired result is obtaied The proof of relatios (6 to ( would ru as above ii Relatios amog two ad four variables fuctios Now we establish N-fractioal relatioships betwee hper-geometric fuctios of two ad four variables These relatios ca be proved o the same lies as adopted i the proof of the relatios i the previous sectio cosiderig lemma ad the defiitios of the hper-geometric fuctios durig the proof Thus usig (6 ad (7 we obtai the followig formulas: { ; ; w µ ; w λ [ w] = ; ; ( B λ λ µ µ { µ ( ( w (- ; ; ; w λ [ ; w] = B λ µ µ ; (5 { λ; µ η ; ( ( w (- ; w = B 6 ; λ ; µ η w (6 { λ ; µ η ; w ; w = B 8 λ ; λ ; µ η w (7
6 athematics Letters 06; (6: { µ ( (-w ; ; ; w λ σ σ λ µ µ = B 5 λ ; ; w (8 { µ ( ( w ; ; ; w λ σ σ µ µ = B 6 λ ; ; w (9 { µ η ( ( w ( ( w ; ; ; w = ; ; w (50 B µ η { λ σ ; ; w w ; w µ ( ( [ ; w ] = 5 ; (5 B σ λ λ µ µ ; ; ; ; w µ λ σ w = B λ λ σ ; µ µ w (5 ; ; ; w λ σ µ η ( w w ( w ( ( w = B A λ σ ; ; µ η w (5 ; ; ; ; w λ σ δ µ ( w = B λ σ δ ; µ µ w (5 7 ; w iii Relatios amog three ad four variables fuctios I this sectio we derive a umber of relatioships ; ; µ η ( ( w w = B λ λ ; µ η w (55 5 ; ( { A λ σ δ ; µ µ µ ; w 6 betwee hper-geometric fuctios of three ad four variables which ca be established o the same lies as i the proof of (5 [ λ λ λ σ δ µ µ µ ; w ] ; ( { w ; ; A λ σ δ µ µ µ A λ σ δ ; µ µ µ ; w (56 (57
7 5 aged Gumaa Bi-Saad: Relatios Amog Certai Geeralied Hper-Geometric uctios Suggested b N-fractioal Calculus ( { B λ λ δ δ δ ; µ ; w A 76! r! r! ( = λ λ δ δ δ ; µ µ µ ; w ( { w ; ; C λ µ µ µ λ λ λ ; µ µ µ ( { w ; ; D λ λ λ µ 60 λ λ λ ; µ µ µ ( { D λ σ σ ; µ ; w 6 λ λ λ σ σ ; µ µ µ { w ; ; E σ δ δ µ µ µ A δ δ σ ; µ µ µ { E λ λ λ δ δ ; µ µ µ ; w 5 λ λ λ δ δ ; µ µ µ w { ; ; E λ λ λ σ µ µ µ λ λ λ σ ; µ µ µ w { λ λ λ δ ; µ η η ; 5 λ λ λ δ ; µ η η { ; ; λ λ λ σ σ µ η η w 7 λ λ λ σ σ ; η µ η w { ; ; λ σ λ µ η η λ λ σ ; η µ η { w ; ; G λ σ δ µ η η σ δ λ ; η η µ { w G λ λ λ σ δ ; µ η η ; λ λ λ σ δ ; η η µ { G λ λ λ σ δ ; µ η η ; w λ λ λ σ δ ; µ η η ; w ; w ; w ; w ; w ; w ; w ; w ; w ; w ; w ; w (58 (59 (60 (6 (6 (6 (6 (65 (66 (67 (68 (69 (70
8 athematics Letters 06; (6: w { K λ σ σ δ ; µ µ µ ; 6 δ σ λ σ ; µ µ µ ; w { K λ λ σ δ σ ; µ µ µ ; w 8 λ λ σ δ σ ; µ µ µ ; w { λ λ σ δ σ ; µ η η ; w λ λ σ δ σ ; η η µ ; w w { λ σ σ δ ; µ η η ; 5 δ σ λ σ ; η µ η ; w { N λ σ δ ε δ ; µ η η ; w 7 δ δ λ σ ε ; µ η η ; w { ; ; N λ σ δ ε δ µ η η w 5 δ δ σ λ ε ; µ µ η ; w w { N λ σ δ ε ; µ η η ; 6 σ λ δ ε ; η µ η ; w w { ; ; P λ σ λ δ µ η η w δ λ σ λ ; µ η η ; w { ; ; P λ λ σ σ δ µ η η w λ λ σ σ δ ; η µ η ; w w { P λ σ σ δ ; µ η η ; λ σ δ σ ; µ η η ; w w { R λ σ δ σ ; µ η η ; 0 λ σ σ δ ; η µ η ; w { ; ; R λ λ σ δ σ µ η η w 0 λ λ σ σ δ ; η µ η { w ; ; S λ σ δ ε µ µ µ 67 ; w λ δ ε σ ; µ µ µ ; w (7 (7 (7 (7 (75 (76 (77 (78 (79 (80 (8 (8 (8
9 55 aged Gumaa Bi-Saad: Relatios Amog Certai Geeralied Hper-Geometric uctios Suggested b N-fractioal Calculus { S λ λ σ δ ε ; µ µ µ ; w 7 λ λ δ ε σ ; µ µ µ ; w { ; ; S λ σ σ δ ε µ µ µ w 7 σ σ λ ε δ ; µ µ µ ; w { T λ λ σ δ σ ; µ µ µ ; w 70 λ λ σ δ σ ; µ µ µ ; w w { T λ σ σ δ ; µ µ µ ; 65 δ σ λ σ ; µ µ µ ; w (8 (85 (86 (87 ultivariable N-fractioal Operator ; Defiitio 5 Let be a multivariable N- otivated b the results of the previous sectios we aim fractioal operator defied b here at presetig a multivariable geeraliatio of the N- fractioal operator (5 as follows ; = { ( = { Γ( ζ = = ( d d C C ( ζ ζ ( ζ ζ (88 πi = ζ where C ad ( Z = Clearl for = (88 reduces to (5 oreover i view of lemma ad the multiomial theorem [ p 9 (0]: m m ( ( = m m m m 0!! = m m (89 the operators ; ca be writte i the form ; ( ( ( [ ; A ; ] (90 where ( A is Lauricella fuctio of -variables (cf [ p (] ad throughout this wor iπ ( ( Γ { Γ( ( A( = e (9 = Net i [] we have established hper-geometric fuctio i several variables ad deoted it b ( K E ( AD which provide multivariable geeraliatio of a umber of ow hper-geometric fuctios The fuctio ( K ( E AD is defied as below
10 athematics Letters 06; (6: ( K ( E [ λ ; ; ] AD = ( λ ( ( m m m m m ( ( ( m! m! (9 m m m m m 0 = m m B the geeral theor of covergece of multiple hpergeometric fuctios (see eg [8 sectio 9]; also [ Chapter 9] it follows that the regio of covergece for ( K ( E AD is ma { < = Now we cosider ol four iterestig multivariable applicatios of the operator (88 Cosiderig ( ; I ; = ; c ( ( O epressig ( i series form [] c ad emploig (89 ad the Legedre duplicatio formula [ p ]: m m = λ λ m m ( λ ( ( we fid I = m m m m m m m m m 0 = ( ( ( m ( (!! m m m m m! m! ; m { = which o usig lemma gives us the desired result: ; ( ; ; c ( ( ( ( ( ( [ ] = A H ; ; (9 ( ( or the other three Lauricella fuctios A B ad lemma ] similarl ields the followig results: ( D [ p ] the operator (88 [ i couctio with (89 ad ; ( A λ λ ; ; σ σ ( [ λ λ ; σ σ ; ] ( = A( A ; ( ; - ; B λ λ ( ( ( ( ( ( [ λ λ ; ; ] ; (9 = A H (95 ; ( ; ; c ( ( ( ( ( ( [ ; ; ] = A H (96 where ( ( H ad ( ( H are Eto s geeralied Hor s fuctios [8 p 97 (5] ; ( ; D ; ; λ ( (
11 57 aged Gumaa Bi-Saad: Relatios Amog Certai Geeralied Hper-Geometric uctios Suggested b N-fractioal Calculus ( ( K ( = A( [ EAD ; λ ; ] (97 iall let us stress that the schema suggested i Sectios ad ca be applied to fid N-fractioal relatios for other geeralied hpergeometric fuctios I a forthcomig papers we will cosider the problems of establishig N- fractioal relatios for other geeralied hpergeometric fuctios b followig the techique discussed i this paper Refereces [] Appell P Sur les séries hpérgeométriques de deu variables et sur des équatios différetielles liéaires au déivées partielles C R Acad Sci Paris 90 ( [] aged G Bi-Saad aisoo A Hussei Operatioal images ad relatios of two ad three variable hpergeometric series Volume Issue J Prog Res ath (JPR [] Bi-Saad aged G A ew multiple hper-geometric ( ( fuctio related to Lauricella s A ad D (commuicated for publicatio [] Chhaa Sharma ad C L Parihar Hpergeometric fuctios of four variables (I J Idia Acad ath Vol No (989 - [5] Che P Srivastava H (997 ractioal calculus operators ad their applicatios ivolvig power fuctios ad summatio of series Appl ath Comput [6] Goal S P Jai R Gaur N (99 ractioal itegral operators ivolvig a product of geeralied hpergeometric fuctios ad a geeral class of polomials II Idia J Pure Appl ath -8 [7] Goal S P Jai R Gaur N (99 ractioal itegral operators ivolvig a product of geeralied hpergeometric fuctios ad a geeral class of polomials Idia J Pure Appl ath 0- [8] Eto H ultiple hpergeometric fuctios ad applicatios Ellis Horwood Ltd Chichester New Yor 976 [9] Eto H Hpergeometric fuctios of three variables J Idia Acad ath Vol (98-9 [0] Hor J Hpergeometrische utioe weier veraderliche ath A 05 ( [] Kalla S L (970 Itegral operators of fractioal itegratio at Notae 89-9 [] Kalla S L Saea R K (969 Itegral operators ivolvig hpergeometric fuctios ath Zeitschr08 - [] Kalla S L Saea R K (97 Itegral operators ivolvig hpergeometric fuctios II Uiv Nac Tucuma Rev Ser A -6 [] Kat S Koul C L (99 O fractioal itegral operators J Idia ath Soc (N S [5] Lauricella G Sulle fuioi ipergeometriche a piu variabili Red Circ ath Palermo 7 ( [6] Nishimoto K ractioal Calculus Vol I Descartes Press Co Koriuma Japa 96(98 [7] Nishimoto K ractioal Calculus Vol II Descartes Press Co Koriuma Japa 96 (98 [8] Sara S Hpergeometric fuctios of three variables Gaita 5 ( [9] Samo S G ad A A Kilbas ad O I arichev ractioal Itegrals ad Derivatives Theor ad Applicatios Gordo ad Breach Amsterdam 99 [0] Sharma C Parihar C L Hpergeometric fuctios of four variables (I Idia Acad ath (989 [] Srivastava H ad Karlsso P W ultiple Gaussia hpergeometric series Halsted Press Bristoe Lodo New Yor 985 [] Srivastava H ad aocha H L A treatise o geeratig fuctios Halsted Press New Yor Brisbae ad Toroto 98
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