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

Siqintuya Jin1, Bai-Ni Guo2,*, Feng Qi3,*
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.

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


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.


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

Cite This Article

Jin, S., Guo, B., Qi, F. (2022). Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities. CMES-Computer Modeling in Engineering & Sciences, 132(3), 781–799.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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