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

• Published in: International journal of applied and computational mathematics Vol. 11; no. 4; p. 120
• Main Authors: Özarslan, Mehmet Ali, Elidemir, Ilkay Onbasi
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.08.2025; Springer Nature B.V
• Subjects: Applications of Mathematics; Bivariate analysis; Calculus; Computational Science and Engineering; Derivatives; Fractional calculus; Hermite polynomials; Integrals; Investigations; Laplace transforms; Mathematical and Computational Physics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nuclear Energy; Operations Research/Decision Theory; Operators (mathematics); Original Paper; Parameter modification; Polynomials; Theoretical; Variables; Bivariate orthogonal polynomials; Mittag Leffler functions; 26A33; Primary 33C50; 44A20; Hermite polynomials; Fractional integrals and derivative
• ISSN: 2349-5103; 2199-5796
• DOI: 10.1007/s40819-025-01932-8

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.