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

Abstract

In this paper, we present a novel method for constructing families of two-variable biorthogonal polynomials by utilizing known families of one-variable biorthogonal and orthogonal polynomials. Employing this approach, we define a new class of two-dimensional Hermite–Konhauser (H–K) polynomials and investigate several of their fundamental properties, including biorthogonality, operational formulas, and integral representations. Furthermore, we examine the behavior of these polynomials under Laplace transformations, as well as fractional integral and derivative operators. Corresponding to the introduced polynomials, we define a new class of bivariate H–K Mittag–Leffler functions and derive analogous properties. To develop a new framework in fractional calculus, we introduce two additional parameters and extend our study to the modified two-dimensional Hermite–Konhauser polynomials and bivariate H–K Mittag–Leffler functions. Additionally, we propose an integral ope-rator whose kernel involves the modified bivariate H–K Mittag–Leffler function. We demonstrate that this operator satisfies the semigroup property and establish its left inverse, which corresponds to a fractional derivative operator. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2025.