Combinatorial Identities Related to Degenerate Stirling Numbers of the Second Kind

Dae San Kim^a, Taekyun Kim<sup>b

a</sup> Department of Mathematics, Sogang University, Seoul 04107, Republic of Korea
^b Department of Mathematics, Kwangwoon University, Seoul 01897, Republic of Korea

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.

Keywords: degenerate Stirling numbers of the second kind, degenerate Bernoulli numbers, Hoppe's generalized chain rule, de Moivre's theorem.

Received: May 5, 2025
Revised: June 2, 2025
Accepted: June 28, 2025

DOI: 10.4213/tm4480

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 330, 176–192

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