On the degenerate higher order Frobenius-Euler polynomials

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 λ-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 λ-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.