A note on degenerate derangement polynomials and numbers

TL;DR

This paper explores the properties and interconnections of degenerate derangement polynomials and numbers, deriving explicit formulas, recurrence relations, and linking them to degenerate gamma distributions.

Contribution

It introduces new explicit expressions, identities, and connections between degenerate derangement polynomials, other special polynomials, and degenerate gamma distributions.

Findings

Derived explicit formulas for degenerate derangement polynomials and numbers.

Established recurrence relations and identities involving related special polynomials.

Connected these polynomials and numbers with moments of degenerate gamma distributions.

Abstract

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.