On Type 2 Degenerate Poly-Frobenius-Euler Polynomials

Background: This paper introduces a class of special polynomials called Type 2 degenerate poly-Frobenius-Euler polynomials, defined using the polyexponential function. Motivated by the expanding theory of degenerate versions of classical polynomials, the paper seeks to enrich the mathematical landscape by constructing generalized structures with deeper combinatorial and analytic properties. Methods: The study employs the method of generating functions combined with Cauchy's rule for the product of two series to derive explicit formulas and identities, enabling systematic manipulation of series expansions. From an analytic perspective, the authors utilized the comparison test and principles of uniform convergence to establish that certain integral representations correspond to holomorphic functions. Results: The researchers successfully derived explicit formulas and identities for the Type 2 degenerate poly-Frobenius-Euler polynomials. They established meaningful connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, they introduced the Type 2 degenerate unipoly-poly-Frobenius-Euler polynomials, defined via the unipoly function, and thoroughly investigated their various properties, including behaviors under differentiation and integration. Conclusion: The study significantly advances the theory of degenerate polynomials by constructing new polynomial families, derivation of explicit identities, and establishing analytic properties. It opens new avenues for future research by bridging classical and generalized combinatorial sequences within a robust analytic framework.