A finite sum involving generalized falling factorial polynomials and degenerate Eulerian polynomials

TL;DR

This paper explores finite sums involving generalized falling factorial polynomials and degenerate Eulerian polynomials, expressing them through degenerate special numbers and polynomials, and deriving their generating functions and recurrence relations.

Contribution

It introduces new connections between generalized factorial sums and degenerate special polynomials, providing explicit formulas and recurrence relations.

Findings

Finite sums expressed via degenerate Stirling, Bernoulli, and Frobenius-Euler polynomials.

Generated functions for degenerate Eulerian polynomials derived.

Recurrence relations established for degenerate Eulerian polynomials.

Abstract

The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the degenerate Bernoulli polynomials and the degenerate Frobenius-Euler polynomials. Secondly, we consider the degenerate Eulerian polynomials and deduce the generating function and a recurrence relation for them.