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.
Abhishek TV (Thu,) studied this question.