Analytical Properties of a k-Mittag–Leffler Function Generated by the k-Gamma Function

This paper studies a generalized Mittag–Leffler function, denoted Eα,k , that is built from the k-Gamma function rather thanthe classical Gamma function. The motivation for examining Eα,k separately from its classical counterpart is that k-deformedspecial functions arise naturally in the study of k-fractional calculus, where the parameter k enters the governing differentialand integral operators directly, and a function tailored to that setting is more convenient for subsequent operator-theoretic workthan repeatedly rewriting results for the ordinary Mittag–Leffler function. We show that Eα,k is an entire function and compute its order of growth in terms of α and k. We then prove an exact reduction identity linking Eα,k to the classical Mittag Leffler function Eβ through a simple rescaling of the argument, and we exploit this identity, together with residue calculus, to produce a Mellin–Barnes contour-integral representation of Eα,k . Using known asymptotic results for Eβ , we obtain the asymptotic behaviour of Eα,k for large |z| inside an appropriate sector of the complex plane. In the second part of the paper we turn to an associated k-fractional Riemann–Liouville integral operator, for which we establish Lp-boundedness, identify a compactness criterion on L2 via the Hilbert–Schmidt property, and show that the operator is quasinilpotent with spectrum {0}, in line with the general theory of compact Volterra operators.