Probabilistic generalization of Spivey-type relation for degenerate Bell polynomials

Taekyun Kim; Dae San Kim; Taekyun Kim; Dae San Kim

doi:10.3934/math.20251295

AIMS Mathematics

2025, Volume 10, Issue 12: 29488-29497. doi: 10.3934/math.20251295

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Probabilistic generalization of Spivey-type relation for degenerate Bell polynomials

Taekyun Kim ^{1
,
,},
Dae San Kim ²

1.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2.
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received: 03 July 2025 Revised: 09 December 2025 Accepted: 10 December 2025 Published: 15 December 2025
MSC : 11B73, 11B83

Following Spivey's pivotal discovery of a recurrence relation for Bell numbers, significant research has emerged concerning various generalizations of Bell numbers and polynomials. For example, Kim and Kim established a Spivey-type recurrence relation specifically for degenerate Bell and Dowling polynomials. In this paper, we extend this work by deriving a probabilistic generalization of Spivey-type recurrence relations for both degenerate Bell and degenerate $ r $-Bell polynomials.
- probabilistic degenerate Bell polynomials,
- probabilistic degenerate $ r $-Bell polynomials,
- Spivey-type relation
Citation: Taekyun Kim, Dae San Kim. Probabilistic generalization of Spivey-type relation for degenerate Bell polynomials[J]. AIMS Mathematics, 2025, 10(12): 29488-29497. doi: 10.3934/math.20251295

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Abstract

Following Spivey's pivotal discovery of a recurrence relation for Bell numbers, significant research has emerged concerning various generalizations of Bell numbers and polynomials. For example, Kim and Kim established a Spivey-type recurrence relation specifically for degenerate Bell and Dowling polynomials. In this paper, we extend this work by deriving a probabilistic generalization of Spivey-type recurrence relations for both degenerate Bell and degenerate $ r $-Bell polynomials.

References

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