Fractional Calculus Operator Emerging from the 2D Biorthogonal Hermite Konhauser Polynomials

Abstract: In the present paper, we introduce a method to construct two variable biorthogonal polynomial families with the help of one variable biorthogonal and orthogonal polynomial families. By using this new technique, we define 2D Hermite Konhauser polynomials and we investigate several properties of them such as biorthogonality property, operational formula and integral representation. We further inverstigate their images under the Laplace transformations, fractional integral and derivative operators. Corresponding to these polynomials, we define the new type bivariate Hermite Konhauser Mittag Leffler function and obtain the similar properties for them. In order to establish new fractional calculus, we add two new parameters and consider 2D Hermite Konhauser polynomials and bivariate Hermite Konhauser Mittag Leffler function. We introduce an integral operator containing the modified bivariate Hermite Konhauser Mittag Leffler function in the kernel. We show that it satisify the semigroup property and obtain its left inverse operator, which corresponds to its fractional derivative operator.