Two unified families of bivariate 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 con-tinuations). We also demonstrate how these functions and operators arise naturally from some fractional partial integro-differential equations of Riemann-Liouville type.(c) 2022 Elsevier Inc. All rights reserved.