Certain Sequences Involving Product of k-bessel Function

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1 It. J. Appl. Coput. Math 018 4:101 ORIGINAL PAPER Certai Sequeces Ivolvig Product of k-bessel Fuctio M. Chad 1 P. Agarwal Z. Haouch 3 Spriger Idia Private Ltd. part of Spriger Nature 018 Abstract Recetly operatioal techiques have draw the attetio of several researchers i the study of geeratig relatios ad suatio forulae. I the preset paper here we itroduce a ew sequece of fuctios ivolvig the product of the geeralized k-bessel fuctio. By usig the operatioal techiques soe geeratig relatios ad fiite suatio forulae of the sequece preseted here are also established. Keywords Special fuctio Geeratig relatios Geeralized k-bessel fuctio Sequece of fuctio Fiite suatio forula Itroductio ad Preliiaries Bessel fuctios first defied by the atheaticia Daiel Beroulli ad the geeralized by Friedrich Bessel are iportat special fuctios ad these are widely used i physics ad egieerig such as Electroagetic waves Heat coductio rotatioal flows sigal processig Diffusio probles Dyaics of floatig bodies etc. Therefore these are of iterest to egieers ad physicists as well as atheaticias. I this paper we ai to itroduce a ew sequece of fuctios ivolvig the product of the geeralized k-bessel fuctio to establish the geeratig relatios ad suatio forulae by usig the operatioal techiques. B Z. Haouch haouch.zakia@gail.co M. Chad ehar.jalladhra@gail.co P. Agarwal goyal.pravee011@gail.co 1 Departet of Matheatics Fateh College for Woe Bathida Idia Departet of Matheatics Aad Iteratioal College of Egieerig Jaipur Idia 3 E3MI Departet of Matheatics FSTE Moulay Isail Uiversity BP Errachidia Morocco

2 101 Page of 9 It. J. Appl. Coput. Math 018 4:101 Recetly Roero et al. 8] see also 1] itroduced the k-bessel fuctio of the first kid for λ γ ν C k R ad Rλ > 0 Rν > 0 as follows: γ λ J kμ z = =0 γ k 1 z 1 Γ k λ + μ + 1! where γ k ad Γ k γ are k-pochheer sybol ad k-gaa fuctio. These are itroduced by Diaz ad Parigua 3] ad defied as: Γk γ +k γ k := Γ k γ k R; γ C \0 γγ + k...γ + 1k N; γ C They gave the relatio with the classical Euler s gaa fuctiosee 8] as: Γ k γ = k γ γ k 1 Γ 3 k whe k = 1 reduces to the classical Pochhaer sybol ad Euler s gaa fuctio respectively see 6]. I ters of the k-pochhaer sybol γ k defied by we itroduce ore geeralized for of k-bessel fuctio ω γλ kνbc ω γλ kνbc z = z as follows: =0 1 c γ z ν+ k Γ k ν + λ + b+1! 4 where λ γ ν c b C ad Rλ > 0 Rν > 0. A ew sequece of fuctio V γ i λ i ; ;α ; k 1...k r s i this paper as: V γ i λ i ; ;α ; k 1...k r s = 1! α ] α =0 ] is itroduced where s + D D d dx ad s are costats k 1...k r are fiite ad oegative iteger p ki is a polyoial i of degree k i where i = 1...radω γλ μνbc is a geeralized k-bessel fuctio which is defied i 4. is based o the work of Mittal 4] Patil ad Thakare 5] Srivastava ad Sigh 9]. For our ivestigatio the followig operatioal techiques are required: exp t exp t uv = β f = β 1 t β+s 5 1 t 1/ 6 f α f = α α+s 1 + t 1+ f 1 + t 1/ 7 v T 1 u D1 + + D1 + + D D D β 1 β = β 1 9

3 It. J. Appl. Coput. Math 018 4:101 Page 3 of t α = 1 t β α β t. 10! Geeratig Relatios I this sectio we stablish here soe geeratig relatio ivolvig the product of geeralized k-bessel fuctio by eployig the operatioal techiques. Theore 1 Let λ i γ i ν i b i c i C; α μ i R; ad s are costat; such that Rλ > 0 Rν > 0 Rα + s > 0 > 0 the we have the followig forula: =0 V γ i λ i ; ;α ; k 1...k r s t = 1 t α+s r pki ] r pki 1 t 1/ ]. where p ki is a polyoial i of degree k i.k i i = 1...r are fiite ad o-egative itegers. Proof To prove the result i Eq. 11 we start fro ew equatio of fuctio give i Eq. 5 fro this equatio we have: =0 V γ i λ i ; ;α ; k 1...k r s t = α pki ] expt α μ i ν i b i c pk1 i ] eployig the operatioal techique give i Eq. 6 the above Eq. 1 reduces to: =0 V γ i λ i ; ;α ; k 1...k r s t = 1 t α+s r pki ] r after replacig t by t i Eq. 13 we have the desired result pki 1 t 1/ ] Theore Let λ i γ i ν i b i c i C; α μ i R; ad s are costat; such that Rλ > 0 Rν > 0 Rα + s > 0 > 0 the we have the followig forula: =0 V γ i λ i ; ;α ; k 1...k r s t = 1 + t α+s 1+ r pki ] r pki 1 + t 1/ ]. where p ki is a polyoial i of degree k i.k i i = 1...r are fiite ad o-egative itegers. 14

4 101 Page 4 of 9 It. J. Appl. Coput. Math 018 4:101 Proof Agai fro Eq. 5 we have: =0 V γ i λ i ; ;α ; k 1...k r s t = α ] exp t α ] 15 applyig the operatioal techique give i Eq. 7 the above Eq. 15 reduces to =0 V γ i λ i ; ;α ; k 1...k r s t = 1 + t α+s 1 pki ] r pki 1 + t 1/ ] 16 which is desired. Theore 3 Let λ i γ i ν i b i c i C; α μ i R; ad s are costat; such that Rλ > 0 Rν > 0 Rα + s > 0 > 0 the we have the followig forula: + V γ i λ i ; ;α + ; k 1...k r s t α+s = 1 t V γ i λ i ; ;α ωγ i λ i ωγ i λ i pki ] 1 t 1/ ] 1 t 1/ ; k 1...k r s. where p ki is a polyoial i of degree k i.k i i = 1...r are fiite ad o-egative itegers. Proof To obtaied the result 17 we ca write Eq. 5as: α ] ] ultiplyig both sides of the above Eq. 18 byexp t exp t =! exp t 17 =! α V γ i λ i ; ;α ; k 1...k r s ] α ωγ i λ i wehave ] ] α V γ i λ i ; ;α ; k 1...k r s ] ωγ i λ i 18 19

5 It. J. Appl. Coput. Math 018 4:101 Page 5 of t! =! exp + α t ] α V γi λi ;μi νi bi ci ;α ; k 1...k r s ] ωγ i λ i eployig the operatioal techique 6 the above Eq. 0 ca be writte as: t + α μ! i ν i b i c pki i ] ] =! α 1 t α+s ωγ i λ i ow usig Eq. 19 i the above Eq. 1 we have: t +!!! 0 V γ i λ i ; ;α 1 t 1/ ; k 1...k r s p ki 1 t ] 1/ α V γ i λ i ; ;α + ; k 1...k r s ] ωγ i λ i = α 1 t α+ s V γ i λ i ; ;α 1 t 1/ ; k 1...k r s p ki 1 t ] 1/ therefore we ca write the above Eq. as: + ωγ i λ i ωγ i λ i V γ i λ i ; ;α + ; k 1...k r s t = 1 t α+ s ] V γ i λ i ; ;α ωγ i λ i 1 1 t 1/ ; k1...k r s p ki 1 t 1/ ] 3 replacig t by t i above Eq. 3 this gives the required result 17. Fiite Suatio Forulas Theore 4 Let λ i γ i ν i b i c i C; α μ i R; ad s are costat; such that Rλ > 0 Rν > 0 Rα + s > 0 > 0 the we have the followig forula: V γ i λ i ; ;α ; k 1...k r s 1 = α! V γ i λ i ; ;0 ; k 1...k r s. where p ki is a polyoial i of degree k i.k i i = 1...r are fiite ad o-egative itegers. 4

6 101 Page 6 of 9 It. J. Appl. Coput. Math 018 4:101 Proof The Eq. 5 ca be writte as: V γ i λ i ; ;α ; k 1...k r s = 1! α ] ow applyig the operatioal techique 8 we have: V γ i λ i ; ;α α 1 ; k 1...k r s = 1! α ] T 1 α 1 ] 5 pki ] = 1! α pki ]!!! s + Ds + + Ds + + D...s D] ] 1 + D1 + + D1 + + D D] α 1 usig the result give i Eq. 9 the above Eq. 6 reduces to the followig for: V γ i λ i ; ;α ; k 1...k r s = 1 i=0 s + i + D pki ] Puttig α = 0 ad replacig by i 6 we get: ] 1!! α. 6 7 V γ i λ i ; ;0 ; k 1...k r s 1 = μ! i ν i b i c pki i ] 1! ] ] = V γ i λ i ; ;0 ; k 1...k r s ] ωγ i λ i 8 9

7 It. J. Appl. Coput. Math 018 4:101 Page 7 of the above Eq. 9gives: 1 1 s + i + D! i=0 ] = V γ i λ i ; ;0 ; k 1...k r s ] ωγ i λ i fro the Eqs. 7ad30 we have the desired result. 30 Theore 5 Let λ i γ i ν i b i c i C; α μ i R; ad s are costat; such that Rλ > 0 Rν > 0 Rα + s > 0 > 0 the we have the followig forula: V γ i λ i ; ;α ; k 1...k r s 1 = α β V γ i λ i ; ;β ; k 1...k r s.! where p ki is a polyoial i of degree k i.k i i = 1...r are fiite ad o-egative itegers. Proof Begis fro Eq. 5 which ca be writte as: 31 =0 V γ i λ i ; ;α ; k 1...k r s t = α ] exp t α ] applyig the operatioal techique give i Eq. 6 the Eq. 3 reduced to: =0 V γ i λ i ; ;α ; k 1...k r s t = 1 t α+ s r p ki 1 t 1/ ] pki ] r 3 33 applyig the result 10; the Eq. 33gives: =0 V γ i λ i ; ;α ; k 1...k r s t α β p k 1 t ] 1/ = 1 t β+ s = ω γλ μνbc α β t! β t! pki ] ] exp t

8 101 Page 8 of 9 It. J. Appl. Coput. Math 018 4:101 = = β =0 β =0 β ] α β t +!! pki ] α β β t!! β pki ] Now equatig the coefficiet of t weget: ] ]. 34 V γ i λ i ; ;α ; k 1...k r s = α β!! β β pki ] ] eployig the result 5 ieq.35 we have the desired forula Cocludig Rearks 1. If we choose b = c = 1 the geeralized k-bessel fuctio reduced to the followig for: z 4 z ω γλ z μ kμ11 z = =0 1 γ k Γ k λ + μ + 1! = z μ J γ λ kμ 36 where λ γ μ C ad Rλ > 0 Rμ > 0. All the results i Sectio reduced to γ λ ivolvig the product of J kμ... If we choose b = 1 c = 1 the geeralized k-bessel fuctio reduced to the k-wright fuctio 7] associated with the followig relatio: ω γλ z μ kμ 11 z = 1 γ k Γ =0 k λ + μ z 4! = z μ W γ kλμ z where λ γ μ C ad Rλ > 0 Rμ > 0. All the results i Sectio reduced to the ivolvig the product of W γ kλμ. z I this way with the help of our ai sequece forula soe geeratig relatios ad fiite suatio forula of the sequece are also established i the preset paper. 37

9 It. J. Appl. Coput. Math 018 4:101 Page 9 of A ew sequece of fuctios is iportat due to presece of geeralized k-bessel fuctio ω γλ kνbc z. O accout of the ost geeral ature of the geeralized k-bessel fuctio ω γλ kνbc z a large uber of sequeces geeratig relatios ad suatio forulae ivolvig sipler fuctios ca be easily obtaied as their special cases by assigig the values to the paraeters. Refereces 1. Cerutti R.A.: O the k-bessel fuctios. It. Math. Foru Choi J. Kuar D.: Solutios of geeralized fractioal kietic equatios ivolvig Aleph fuctios. Math. Cou Diaz R. Parigua E.: O hypergeoetric fuctios ad k-pochhaer sybol. Divulg. Math Mittal H.B.: Biliear ad Bilateral geeratig relatios. A. J. Math Patil K.R. Thakare N.K.: Operatioal forulas for a fuctio defied by a geeralized Rodrigues forula- II. Sci. J. Shivaji Uiv Raiville E.D.: Special Fuctios. Macilla New York Roero L. Cerutti R.: Fractioal calculus of a k-wright type fuctio. It. J. Cotep. Math. Sci Roero L.G. Dorrego G.A. Cerutti R.A.: The k-bessel fuctio of first kid. It. Math. Foru Srivastava A.N. Sigh S.N.: Soe geeratig relatios coected with a fuctio defied by a Geeralized Rodrigues forula. Idia J. Pure Appl. Math

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