On a generalization of Mittag-Leffler function and its properties

[摘要] Let s and z be complex variables, Gamma(s) the Gamma function, and (s)(v) = Gamma(s+v)/Gamma(s) for any complex v the F(s) generalized Pochhammer symbol. The principal aim of the paper is to investigate the function [GRAPHICS] where alpha, beta, gamma is an element of C; Re(alpha) > 0, Re(beta) > 0, Re(gamma) > 0 and q is an element of (0, 1) boolean OR N. This is a generalization of the exponential function exp(z), the confluent hypergeometrie function Phi(gamma, alpha; z), the Mittag-Leffler function E-alpha(z), the Wiman's function E-alpha,E-beta(z) and the function E-alpha,beta(gamma)(z) defined by Prabhakar. For the function E-alpha,beta(gamma,q)(z) its various properties including usual differentiation and integration, Laplace transforms, Euler (Beta) transforms, Mellin transforms, Whittaker transforms, generalised hypergeometric series form, Mellin-Bames integral representation with their several special cases are obtained and its relationship with Laguerre polynomials, Fox H-function and Wright hypergeometric function is also established. (c) 2007 Elsevier Inc. All rights reserved.

[发布日期] 2007-12-15

[关键词] confluent hypergeometric function;Euler transform;fox h-function;laplace transform;mellin transform;mittag-leffler function;whittaker transform;wiman's function;wright hypergeometric function