Unified Wright-Mittag-Leffler generalizations of the two-variable Hermite polynomials with applications

M.A. Pathan; Talha Usman

doi:10.24193/subbmath.2026.3.2

Authors

M.A. Pathan Department of Mathematics, Aligarh Muslim University, Aligarh, India https://orcid.org/0000-0003-3918-7901
Talha Usman Preparatory Studies Center, Mathematics and Computing Skills Unit, University of Technology and Applied Sciences, Sur, Oman

DOI:

https://doi.org/10.24193/subbmath.2026.3.2

Keywords:

Special Functions, Operational Calculus

Abstract

We develop a unified Wright-type generalization of the two-variable Hermite--Kampé de Fériet polynomials by introducing generating kernels built from Wright and Mittag--Leffler functions. Two normalizations Wright--Bessel and Mittag--Leffler are treated simultaneously, yielding explicit coefficient formulas, Appell--Sheffer-type lowering and raising identities, Euler homogeneity, Rodrigues representations and discrete forward-difference analogues. The Mittag--Leffler normalization recovers the full classical Hermite structure, while the Wright--Bessel case produces deformed convolution laws. As an application, we show that these generalized polynomials generate explicit solutions to Caputo fractional heat-type equations. The framework unifies continuous, fractional and discrete Hermite-type structures within a single analytic scheme.

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