Bivariate discrete Mittag-Leffler functions with associated discrete fractional operators

Abstract

Based on the recently published article about bivariate Mittag-Leffler function Eα,β,γδ(x) by Fernandez et al. (2020), we introduce the bivariate discrete Mittag-Leffler function, denoted by Eα,β,γ¯δ(λ1,λ2;x), as a discrete version of the results of Fernandez et al. under some constraints in this study. We establish this new definition to find a fractional difference equation. Then, we employ the fractional sum and differences formulas to get the results with respect to the bivariate discrete Mittag-Leffler function. Moreover, we give the discrete Laplace transform of the corresponding discrete function and build up the discrete sum operator to show up the semigroup property on some constraints. Also, left inverse of the discrete sum operator is given. Finally, we end the paper by two examples and conclusion.