Abstract
In this paper, we introduce and study a Mittag-Leffler-type function of two variables E1 (x, y) and a generalization of Mittag-Leffler-type function of one variable as limiting case of E1 (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 E1 (x, y), considering all possible cases. Next, we give two differential equations for E1 (x, y) and one differential equation for for some particular values of the parameters. We further obtain two integral representations and Mellin–Barnes contour integral representation of E1 (x, y). We also obtain the Laplace transform of one and two dimensions of E1 (x, y) and its fractional integral and derivative. Next, we define an integral operator with E1 (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.
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