Certain properties and numerical applications of generalized hybrid special polynomials associated with Hermite polynomials

Abstract

This paper introduces a novel class of generalized Hermite-based Apostol-type FrobeniusEuler polynomials and numbers of order $\nu$ and level $\alpha$. We establish fundamental identities and properties by employing generating function techniques, including summation formulas, differential and integral relations, and addition theorems. Furthermore, we investigate their connections with Stirling numbers of the second kind and various other polynomial families. Additionally, we derive a corresponding differential equation and a recurrence relation for these newly defined polynomials. To visualize their behaviour, we utilize Maple to compute numerical values and illustrate the distribution of their zeros through surface plots.