SOME INTEGRAL TRANSFORMS OF THE GENERALIZED k-MITTAG-LEFFLER FUNCTION

In the paper, the authors generalize the notion “k-Mittag-Leffler function”, establish some integral transforms of the generalized k-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized k-Wright function. 1. Preliminaries Throughout this paper, let C, R, R0 , R, Z − 0 , and N denote respectively the sets of complex numbers, real numbers, non-negative numbers, positive numbers, non-positive integers, and positive integers. The Pochhammer symbol (λ)ν can be defined for λ, ν ∈ C by (λ)ν = Γ(λ+ν) Γ(λ) , where Γ(z) = lim n→∞ n!n ∏n k=0(z + k) , z ∈ C \ Z0 is called the classical gamma function and its reciprocal 1 Γ is analytic on the whole complex plane C. See [14, Chapter 5], [16, Section 1], and [24, Section 1.1]. In particular, when ν ∈ {0} ∪ N, the quantity (λ)n = { 1, ν = 0 λ(λ+ 1) · · · (λ+ n− 1), n ∈ N is called the rising factorial. See [17] and closely-related references therein. E-mail addresses: [email protected], [email protected], [email protected], [email protected], [email protected]. 2010 Mathematics Subject Classification. Primary 33E12; Secondary 33C20, 44A20, 44A30, 65R10.