This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields.

The primary objective of this research is to introduce and investigate novel polynomial variants termed ${\Delta }_h$ Laguerre polynomials. This unique polynomial type integrates the monomiality principle alongside operational rules. Through this innovative approach, the study delves into uncharted territory, unveiling fresh insights that build upon prior research endeavours. Notably, the ${\Delta }_h$ Laguerre polynomials exhibit significant utility in the realm of quantum mechanics, particularly in the modelling of entropy within quantum systems. The research meticulously unveils explicit formulas and elucidates the fundamental properties of these polynomials, thereby forging connections with established polynomial categories. By shedding light on the distinct characteristics and functionalities of the ${\Delta }_h$ Laguerre polynomials, this study contributes significantly to their comprehension and application across diverse mathematical and scientific domains.

The fundamental aim of this paper is to introduce the concept of poly-Jindalrae and poly-Gaenari numbers and polynomials within the context of degenerate functions. Furthermore, we give explicit expressions for these polynomial sequences and establish combinatorial identities that incorporate these polynomials. This includes the derivation of Dobinski-like formulas, recurrence relations, and other related aspects. Additionally, we present novel explicit expressions and identities of unipoly polynomials that are closely linked to some special numbers and polynomials.