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The present paper deals with the study of a generalized Mittag-Leffler function and associated fractional operator. The operator has been discussed in the space of Lebesgue measurable functions. The composition with Riemann–Liouville fractional integration operator has been obtained.

The aim of this paper is to give some convergence conditions of the <em>_pR_q(α; β; z)</em> function. We also derive the integral representation of the function <em>_pR_q(α; β; z)</em> in the form of Mellin-Barnes Integral including its analytic property.

UDC 517.5 The paper deals with solving the integral equation with a generalized Mittag-Leffler function E α , β γ , q ( z ) that defines a kernel using a fractional integral operator. The existence of the solution is justified and necessary conditions on the integral equation admiting a solution are discussed. Also, the solution of the integral equation is derived.