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arXiv · 2509.03415

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

Taekyun Kim·Dae san Kim·Kyo-Shin Hwang·Dmitry V. Dolgy
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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.

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Taekyun Kim, Dae san Kim, Kyo-Shin Hwang, Dmitry V. Dolgy. 2025-09-03. A note on new type degenerate Srirling numbers of the first kind. https://arxiv.org/abs/2509.03415

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