@article{Zayed2025, 
author = {Mohra Zayed and Maged G. Bin-Saad and Waleed K. Mohammed},
title = {On Mittag-Leffler-Gegenbauer polynomials arising by the convolution of Mittag-Leffler function and Hermite polynomials},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {7},
pages = {16642-16663},
keywords = {Mittag-Leffer function, Gegenbauer polynomials, Hermite polynomials, recurrence relations, integrals, fractional calculus, kinetic equation, orthogonality, symbolic operator},
url = {https://www.sciopen.com/article/10.3934/math.2025746},
doi = {10.3934/math.2025746},
abstract = {Gegenbauer polynomials hold a significant role in the constructive theory of spherical functions, while the Mittag-Leffler function is widely used in fractional calculus. In this paper, we introduce a new class of Mittag-Leffler-Gegenbauer polynomials (MLGPs) by convolutionally combining the classical Hermite polynomials with the Mittag-Leffler function of three parameters. We explore some of its aspects, such as symbolical identities, recurrence relations, differential equations, generating functions, integral representations, finite summations, and Rodrigues-type and orthogonal formulas. Additionally, we demonstrate the relevance of the MLGPs by developing and solving a fractional kinetic equation associated with the MLGPs in the kernel. Finally, employing Saigo fractional-type operators, we establish fractional integrals and derivatives formulae for our innovative MLGPs. We conclude by proposing an open question regarding the Hermite numbers and their umbral calculus for further discussion in the field of this study.}
}