A note on new type degenerate Srirling numbers of the first kind

2025/09/03 by Taekyun Kim, Kim, Taekyun, Dae san Kim +5
Mathematics · #11B73 #11B83 #60-08 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.03415

openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new sequence of unsigned degenerate Stirling numbers of the first kind. Following the work of Adell-Lekuona, who represented unsigned Stirling numbers of the first kind as multiples of the expectations of specific random variables, we express our new numbers as finite sums of multiples of the expectations of certain random variables. We also provide a representation of these new numbers as finite sums involving the classical unsigned Stirling numbers of the first kind. As an inversion formula, we define a corresponding sequence of new type degenerate Stirling numbers of the second kind. We derive expressions for these numbers as finite sums that involve the Stirling numbers of the second kind.

Citations

• Probabilistic degenerate poly-Bell polynomials associated with random variables
• Probabilistic degenerate derangement polynomials

Related

• Some identities and properties on degenerate Stirling numbers
• A generalization of Stirling numbers
• A probabilistic generalization of the Stirling numbers of the second\n kind
• Some identities involving degenerate r-Stirling numbers
• A note on degenerate Stirling numbers of the first kind