Bivariate Mittag-Leffler functions arising in the solutions of convolution integral equation with 2D-Laguerre–Konhauser polynomials in the kernel

Recently, the 2D-Laguerre–Konhauser polynomials were introduced in [1]. In the present paper, first of all, we propose another bivariate polynomial family which is bi-orthonormal with the 2D-Laguerre–Konhauser polynomials. Then, we consider a convolution integral equation with 2D-Laguerre–Konhauser polynomials in the kernel and we obtain its solution by introducing a new family of bivariate Mittag-Leffler functions. Furthermore, we introduce a double (fractional) integral operator including bivariate Mittag-Leffler functions in the kernel. This integral operator includes the double Riemann–Liouville fractional integral operator. We investigate its transformation properties on the continuous function and Lebesgue summable function spaces. Also, the semigroup property of the operator is investigated. We further study some miscelenenous properties of 2D-Laguerre–Konhauser polynomials and bivariate Mittag-Leffler functions such as generating function, Schläfli’s integral represantation. Finally, we approximate to the image of any bivariate continuous function under the action of the proposed double integral operators.