Open Access

ARTICLE

Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

Siqintuya Jin¹, Bai-Ni Guo^2,*, Feng Qi^3,*
1 College of Mathematics and Physics, Inner Mongolia Minzu University, Tongliao, 028043, China
2 School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, China
3 School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China
* Corresponding Authors: Bai-Ni Guo. Email: ; Feng Qi. Email:
(This article belongs to this Special Issue: Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)

Computer Modeling in Engineering & Sciences 2022, 132(3), 781-799. https://doi.org/10.32604/cmes.2022.019941

Received 25 October 2021; Accepted 24 January 2022; Issue published 27 June 2022

Abstract

In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, important, useful, and applicable in combinatorial number theory.

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Keywords

Connection; equivalence; closed-form formula; combinatorial identity; partial Bell polynomial; falling factorial; rising factorial; binomial coefficient; Stirling number of the first kind; Stirling number of the second kind; problem

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