A Mittag-Leffler-type function of two variables

In this paper, we introduce and study a Mittag-Leffler-type function of two variables E-1 (x, y) and a generalization of Mittag-Leffler-type function of one variable E-alpha 2,delta 1;alpha 3,delta 2(gamma 1,alpha 1) (x) as limiting case of E-1 (x, y), which includes several Mittag-Leffler-type functions of one variable as its special cases. Here, we first obtain the domain of convergence of E-1 (x, y), considering all possible cases. Next, we give two differential equations for E-1 (x, y) and one differential equation for E-alpha 2,delta 1;alpha 3,delta 2(gamma 1,alpha 1) (x) for some particular values of the parameters. We further obtain two integral representations and Mellin-Barnes contour integral representation of E-1 (x, y). We also obtain the Laplace transform of one and two dimensions of E-1 (x, y) and its fractional integral and derivative. Next, we define an integral operator with E-1 (x, y) as a kernel and show that it is bounded on the Lebesgue measurable space L(a, b). Finally, we introduce one more Mittag-Leffler-type function of two variables.