Combinatorial identities related to degenerate Stirling numbers of the second kind
Taekyun Kim, Dae san Kim
TL;DR
The paper addresses degenerate Stirling numbers of the second kind ${n \brace k}_{\lambda}$ as the coefficients in $(x)_{n,\lambda} = \sum_{k=0}^{n} {n \brace k}_{\lambda} (x)_k$, extending the classical ${n \brace k}$. It develops properties, recurrence relations, and explicit expressions, and introduces the polynomials $S_n(x,r|\lambda)$, $K_r(x|\lambda)$, $S_{n,r}(x|\lambda)$, and $T_n(x,r|\lambda)$ with representations such as $K_r(x|\lambda) = e^{x} \sum_{j=0}^{r} x^{j} {r \brace j}_{\lambda}$. A degenerate Euler-type formula is established by $\frac{1}{k!}(e_{\lambda}(t)-1)^{k} = \sum_{n=k}^{\infty} {n \brace k}_{\lambda} \frac{t^{n}}{n!}$, and the degenerate Bernoulli numbers $\beta_{n,\lambda}^{(\alpha)}$ are expressed in terms of ${n+j \brace j}_{\lambda}$, linking to probabilistic and umbral interpretations. The results provide explicit combinatorial identities and generating-function techniques for degenerate structures, with potential applications to degenerate variants of Euler/ Bernoulli frameworks and their operator representations.
Abstract
The study of degenerate versions of certain special polynomials and numbers, which was initiated by Carlitz's work on degenerate Euler and degenerate Bernoulli polynomials, has recently seen renewed interest among mathematicians. The aim of this paper is to study some properties, certain identities, recurrence relations and explicit expressions for degenerate Stirling numbers of the second kind, which are a degenerate version of the Stirling numbers of the second kind. These numbers appear very frequently when we study various degenerate versions of many special polynomials and numbers. Especially, we consider some closely related polynomials and power series in connection with a degenerate version of Euler's formula for the Stirling numbers of the second kind.