On a certain bivariate Mittag‐Leffler function analysed from a fractional‐calculus point of view

Mittag‐Leffler functions of one variable play a vital role in several areas of study. Their connections with fractional calculus enable many physical processes, such as diffusion and viscoelasticity, to be efficiently modelled. Here, we consider a Mittag‐Leffler function of two variables and the associated double integral operator, with the goal of establishing once again connections with fractional calculus. By working from the fractional‐calculus viewpoint, it is possible to obtain many new results concerning the double integral operator, including a series formula and a bivariate chain rule. We also discover a left inverse operator, which completes this model of fractional calculus. As applications, we solve some initial value problems and use a modified Stancu‐Bernstein model to approximate the image of a Hölder‐continuous function under the action of our double integral operator.