A new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type

In this paper, we consider a new class of degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type, denoted by $ \mathcal{W}^{(\alpha)}_{n, \lambda}(\delta, \zeta; \rho; \mu) $. We obtain several summation formulae, a recurrence relation, two difference operator formulas, two derivative operator formulas, an implicit summation formula, and a symmetric property for these polynomials. Also, we provide a representation of the degenerate differential operator on the degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type. Moreover, we define the degenerate unified Hermite-based Apostol-Stirling polynomials of the second kind and derive some properties of these newly established polynomials. In addition, we prove multifarious correlations, including the new polynomials. Furthermore, we list the first few degenerate unified Bernoulli-Euler Hermite polynomials of Apostol type for some special cases and present data visualizations of zeros forming 2D and 3D structures. Finally, we provide a table covering approximate solutions for the zeros of $ \mathcal{W}^{(\alpha)}_{n, 3}(\delta, 4;3;2) $.