On a generalization of Mittag-Leffler function and its properties

Abstract

Let s and z be complex variables, Γ ( s ) the Gamma function, and ( s ) ν = Γ ( s + ν ) Γ ( s ) for any complex ν the generalized Pochhammer symbol. The principal aim of the paper is to investigate the function E α , β γ , q ( z ) = ∑ n = 0 ∞ ( γ ) q n Γ ( α n + β ) z n n ! , where α , β , γ ∈ C ; Re ( α ) > 0 , Re ( β ) > 0 , Re ( γ ) > 0 and q ∈ ( 0 , 1 ) ∪ N . This is a generalization of the exponential function exp ( z ) , the confluent hypergeometric function Φ ( γ , α ; z ) , the Mittag-Leffler function E α ( z ) , the Wiman's function E α , β ( z ) and the function E α , β γ ( z ) defined by Prabhakar. For the function E α , β γ , q ( z ) its various properties including usual differentiation and integration, Laplace transforms, Euler (Beta) transforms, Mellin transforms, Whittaker transforms, generalised hypergeometric series form, Mellin–Barnes 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.