Unified Wright-Mittag-Leffler generalizations of the two-variable Hermite polynomials with applications
DOI:
https://doi.org/10.24193/subbmath.2026.3.2Keywords:
Wright–Hermite polynomials, Mittag–Leffler functions, two-variable Hermite polynomials, fractional heat equation, Wright functions, generating functionsAbstract
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.
Mathematics Subject Classification (2010): 26A33, 33C10, 33C45, 33E20.
Received 19 April 2026; Accepted 01 June 2026.
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