On r-Bell-Based Apostol-Frobenius-Type Poly-Euler Polynomials

This paper introduces a novel variation of Frobenius-Euler poly- nomials derived from Bell numbers and Apostol-type functions, in- corporating the polylogarithm concept. We explore their structure and properties through various analytical methods, with a particular emphasis on generating functions designed for higher-order Apostol- Frobenius-Type Poly-Euler polynomials based on Bell numbers. These functions facilitate the derivation of both explicit and implicit sum- mation formulas. Furthermore, we establish symmetric identities that unveil intricate polynomial relationships. This integration provides a new framework that deepens the understanding and expands the applicability of Poly-Euler polynomials. Our findings contribute to combinatorial and algebraic mathematics, fostering further research in related areas.