GENERALIZATION OF SPIVEY'S RECURRENCE RELATION

GENERALIZATION OF SPIVEY'S RECURRENCE RELATION

3 Apr 2024 | TAEKYUN KIM AND DAE SAN KIM
This paper generalizes Spivey's recurrence relation for Bell numbers to probabilistic $r$-Bell polynomials associated with a random variable $Y$. The authors define probabilistic $r$-Stirling numbers and probabilistic $r$-Bell polynomials, extending the classical definitions to a probabilistic setting. They derive explicit expressions for these probabilistic extensions in terms of the $n$th moments of $S_{j+r}$ and present the generating function, finite sum expressions, and recurrence relations for the probabilistic $r$-Bell polynomials. The results are then extended to probabilistic degenerate Bell polynomials and their applications, providing a comprehensive framework for understanding these probabilistic extensions and their potential applications in various fields.This paper generalizes Spivey's recurrence relation for Bell numbers to probabilistic $r$-Bell polynomials associated with a random variable $Y$. The authors define probabilistic $r$-Stirling numbers and probabilistic $r$-Bell polynomials, extending the classical definitions to a probabilistic setting. They derive explicit expressions for these probabilistic extensions in terms of the $n$th moments of $S_{j+r}$ and present the generating function, finite sum expressions, and recurrence relations for the probabilistic $r$-Bell polynomials. The results are then extended to probabilistic degenerate Bell polynomials and their applications, providing a comprehensive framework for understanding these probabilistic extensions and their potential applications in various fields.
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Understanding Generalization of Spivey%E2%80%99s Recurrence Relation