Two unified families of bivariate Mittag-Leffler functions.

• Bivariate Mittag-Leffler functions are classified into broad families. • All the studied functions emerge as solutions of fractional integro-differential equations. • Special cases whose importance has already been seen in the literature. • Indications for future extensions to trivariate and multivariate Mittag-Leffler functions. The various bivariate Mittag-Leffler functions existing in the literature are gathered here into two broad families. Several different functions have been proposed in recent years as bivariate versions of the classical Mittag-Leffler function; we seek to unify this field of research by putting a clear structure on it. We use our general bivariate Mittag-Leffler functions to define fractional integral operators (which have a semigroup property) and corresponding fractional derivative operators (which act as left inverses and analytic continuations). We also demonstrate how these functions and operators arise naturally from some fractional partial integro-differential equations of Riemann–Liouville type. [ABSTRACT FROM AUTHOR]