On the degenerate higher order Frobenius-Euler polynomi-als

Abstract

Degenerate versions of special numbers, originating from the work of L. Carlitz, play a crucial role in various fields, including pure and applied mathematics, com- binatorics, number theory, and mathematical physics. They are actively under investigation by numerous researchers. Recently, [D. S. Kim, T. Kim, J. Math. Anal. Appl., 493 (2021), 21 pages] introduced the A-umbral calculus as a research tool specifically for degenerate special polynomials, utilizing it to establish connections between special polynomials and their degenerate counterparts. In this paper, we investigate the relationships between degenerate Frobenius-Euler polynomials and other versions of degenerate special polynomials and numbers. By employing A-umbral calculus, explicit formulas for Frobenius-Euler polynomials of order r are derived. The presented formulas reveal connections between these polynomials and well-known special numbers and polynomials. Additionally, the distribution patterns of the roots of these polynomials are examined.