On Bilateral Generating Relation for a Sequence of Functions

Transcription

1 It. Joural of Math. Aalysis, Vol. 2, 2008, o. 8, O Bilateral Geeratig Relatio for a Sequece of Fuctios A. K. Shukla ad J. C. Prajapati Departmet of Mathematics S.V. Natioal Istitute of Techology, Surat , Idia ajayshukla2@rediffmail.com jyotidra18@rediffmail.com Abstract I this paper, we obtaied a bilateral geeratig relatio for a sequece of fuctio i.e. {V (α,β (x; a, k, s/ = 0, 1, 2, } which is recetly itroduced by Shukla ad Prajapati [3 defied as (1.1 by applyig Srivastava s Theorem [6, ad its special cases have also bee discussed. Mathematics Subject Classificatio: 33E10, 33E12, 33E99, 44A45 Keywords: Mittag-Leffler fuctio, Bilateral geeratig relatios, Kohauser polyomials, Laguerre polyomials 1 Itroductio ad Prelimiaries Recetly, Shukla ad Prajapati [3 itroduced a sequece of fuctio (x; a, k, s} defied as: {V (α,β V (α,β (x; a, k, s = 1! x β E α {p k (x}θ [x β E α { p k (x} (1.1 p k (x is a polyomial i x of degree k ad the differetial operator θ x a (s + xd, D d, a ad s beig costats. dx E α (z is Mittag-Leffler fuctio [1 defied as: E α (z = z Γ(α +1, (1.2 z is a complex variable ad Γ(s is a gamma fuctio, α>0.

2 384 A. K. Shukla ad J. C. Prajapati This sequece of fuctio (1.1 icorporates several polyomial systems. A geeral class of polyomials G (α,β (x, r, p, k defied by Shukla ad Prajapati [4 as: G (α,β (x, r, p, k = 1! x β k E α (px r θ [x β E α ( px r (1.3 E α (zis Mittag-Leffler fuctio which is defied as (1.2. The polyomials G (α,β (x, r, p, k provide a elegat uificatio of the various kow geeralizatios of the classical Hermite, Laguerre ad Kohauser polyomials. This is easy to prove followig relatio by usig (1.1 ad (1.3 as: G (α,β (x, p, k, a = x a V (α,β (x; a, k, 0 (whe p k (x =px k. (1.4 Shukla ad Prajapati [3 have also proved followig liear geeratig relatio: ( + m V (α,β +m (x; a, k, st = (1 ax α β+s m =0, 1, 2,. E α {p k (x} E α [p k {x(1 ax a 1/a } V (α,β {x(1 ax a 1/a ; a, k, s} (1.5 Srivastava [6 itroduced a geeral method of obtaiig biliear, bilateral or mixed multilateral geeratig relatios for all fuctio Sν (x satisfyig A ν,sν+(xt = f (x, {g(x, } ν Sν(h(x, (1.6 ν is a arbitrary complex umber. He has also proved Theorem 3, i which he showed icely how these two geeral theorems are applicable to derive a large variety of biliear, bilateral or mixed multilateral geeratig relatios of several variables. Theorem 2 (Srivastava [6: If F p,μ q,ν [x; y 1,,y s ; t = C μ,ν S ν+q (xω μ+p(y 1,,y s t (1.7

3 O bilateral geeratig relatio 385 ad the fuctios Sμ (x defied by (1.6, Cμ,ν 0,μad ν are arbitrary complex umbers, p ad q are positive itegers, Ω μ (y 1,,y s isa o-vaishig fuctio of s variables i.e. y 1,,y s, s 1. The S ν+(x,q,ν(y 1,,y s ; zt [ { t = f (x, {g(x, } ν Fq,ν p,μ h(x, ; y 1,,y s ; z W,q,ν p,μ (y 1,,y s ; z is a polyomial of degree as:,q,ν (y 1,,y s ; z = k=0 } q (1.8 g(x, [ i z, which is defied q [ q A ν+qk, qk Cμ,ν k Ω μ+pk (y 1,,y s z k. (1.9 Special case: For s = 1, the bilateral geeratig relatios (1.8 ca be writte as: [ { Sν+ p,μ t } q (xy,q,ν (y; zt = f (x, {g(x, } ν G p,μ q,ν h(x, ; y; z g(x, (1.10 ad [ Y,q,ν p,μ (y; z = G p,μ q,ν[x; y; t = q k=0 A ν+qk, qk Cμ,ν k Θ μ+pk (yz k (1.11 C μ,ν S ν+q(xθ μ+p (yt (1.12 C μ,ν 0,Θ μ (y 0 is a arbitrary fuctio of y, p ad q are positive itegers, μ ad ν are arbitrary complex umbers. Kohauser polyomials of secod kid (Srivastava [7: It is deoted by the symbol Z α (x; k ad defied as, ( ( 1 j j Z α Γ(k + α +1 (x; k =! x kj Γ(kj + α +1 (1.13

4 386 A. K. Shukla ad J. C. Prajapati k is a positive iteger. L α,β (x Polyomials (Prabhakar ad Suma [2: It is defied as: L α,β (x = Γ(α + β +1! k=0 ( k x k Γ(αk + β +1k! Re(β > 1 ad α is ay complex umber with Re(α > 0. (1.14 The followig relatios are also established by Prabhakar ad Suma [2 2 Mai Result L 1,β (x = Zβ (x;1 = Lβ (x. (1.15 O comparig (1.5 with (1.6 ad replacig ν by m (a o-egative iteger, we get ( + m A m, =, S m+(x = V (α,β +m (x; a, k, s, g(x, = 1, ( Sm(h(x, = V (α,β x(1 ax a 1/a ; a, k, s, f (x, = (1 ax a β+s E α {p k (x} E α [p k {x(1 ax a 1/a } ad applyig Theorem 2 (Srivastava [6 yields bilateral geeratig relatio for the sequece of fuctio V (α,β (x; a, k, s which is give as: For s 1 V (α,β +m (x; a, k, s,q,m (y 1,,y s ; zt =(1 ax a β+s E α {p k (x} E α [p k {x(1 ax a 1/a } F p,μ q,m{x(1 ax a 1/a ; y 1,,y s ; zt q } (2.1,q,ν (y 1,,y s ; z = [ q A m+qj, qj A j Ω m+pj (y 1,,y s z j = [ q ( + m qj A j Ω m+pj (y 1,,y s z j (2.2

5 O bilateral geeratig relatio 387 ad F p,μ q,m[x; y 1,,y s ; t = A V (α,β m+q(x; a, k, sω μ+p (y 1,,y s t (2.3 A 0,μ is a arbitrary complex umber, p ad q are positive itegers. 3 Applicatios (i If s = 1, the bilateral geeratig relatio (2.1 reduces to, V (α,β +m (x; a, k, sw,q,m p,μ (y; zt = (1 ax a β+s E α {p k (x} E α [p k {x(1 ax a 1/a } Gp,μ q,m {x(1 axa 1/a ; y; zt q } (3.1,q,ν(y; z = [ q A m+qj, qj A j Θ m+pj (yz j = [ q ( + m qj A j Θ m+pj (yz j (3.2 ad G p,μ q,m[x; y; t = A V (α,β m+q(x; a, k, sθ μ+p (yt (3.3 A 0,Θ μ (y 0 is a arbitrary fuctios of y, μ is a arbitrary complex umber, p ad q are positive itegers. ( 1 (ii Choosig m =0,q =1,Ω μ (y 1,,y s 1, A = i (2.1 Γ(γ + δ +1 ad usig (1.14, we get followig ew bilateral geeratig relatio for the sequece of fuctio V (α,β (x; a, k, s as

6 388 A. K. Shukla ad J. C. Prajapati V (α,β (x; a, k, s,1,0 (y 1,,y s ; zt =(1 ax a β+s E α {p k (x} E α [p k {x(1 ax a 1/a } F p,μ 1,0 {x(1 axa 1/a ; y 1,,y s ; zt} (3.4,1,0(y 1,,y s ; z = A j, j A j z j = ( j A j z j = ( j ( 1 j Γ(γ + δ +1 zj =! Γ(γ + δ +1 Lγ,δ (z (3.5 ad F p,μ 1,0 [x; y 1,,y s ; zt = ( 1 Γ(γ + δ +1 V (α,β (x; a, k, s z t (3.6 hece (3.4 becomes! Γ(γ + δ +1 V (α,β (x; a, k, s L γ,δ (zt = (1 ax a β+s E α {p k (x} ( 1 E α [p k {x(1 ax a 1/a } Γ(γ + δ +1 V (α,β {x(1 ax a 1/a ; a, k, s}z t. (3.7 Therefore, we arrived at the coclusio that (2.1, (3.1 ad (3.7 are very iterestig bilateral geeratig relatios ad supposed to be ew to the literature. Refereces [1 Mario N. Berbera e Satos, Properties of the Mittag-Leffler relaxatio fuctio, J. Mathematical Chemistry, 38(4 (2005, [2 T.R. Prabhakar ad Rekha Suma, Some Results o the polyomials L α,β (x, Rocky Moutai J. Math. 8(4 (1978,

7 O bilateral geeratig relatio 389 [3 A.K. Shukla ad J.C. Prajapati, O a class of polyomials defied by a geeralized Rodrigues formula, Commuicated for publicatio. [4 A.K. Shukla ad J.C. Prajapati, Geeralizatio of a class of Polyomials ad its coectio with Mittag-Leffler fuctio, Proceedigs of the Iteratioal Coferece of Society of Special Fuctio ad their Applicatios (SSFA, 7 (2006. [5 A.N. Srivastava ad L.S. Sigh, A ote o a class of polyomials, Idia J. Pure Appl. Math. 14(2 (1983, [6 H.M. Srivastava, Some bilateral geeratig fuctios for a certai class of special fuctios I ad II, Neder. Akad. Wetesch. Proc. Ser. A83= Idag. Math. 42 (1980, [7 H.M. Srivastava, A multiliear geeratig fuctio for the Kohauser set of biorthogoal polyomials, Pacific J. Math. 117(1 (1985, Received: Jauary 29, 2007