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Poly-falling factorial sequences and poly-rising factorial sequences
- Source :
- Open Mathematics, Vol 19, Iss 1, Pp 1620-1635 (2021)
- Publication Year :
- 2021
- Publisher :
- De Gruyter, 2021.
-
Abstract
- In this paper, we introduce generalizations of rising factorials and falling factorials, respectively, and study their relations with the well-known Stirling numbers, Lah numbers, and so on. The first stage is to define poly-falling factorial sequences in terms of the polyexponential functions, reducing them to falling factorials if k=1k=1, necessitating a demonstration of the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences. The later part of the paper deals with poly-rising factorial sequences in terms of the polyexponential functions, reducing them to rising factorial if k=1k=1. We study some relations: between poly-falling factorial sequences and poly-rising factorial sequences; between poly-rising factorial sequences and the Stirling numbers of the first kind and the power of xx; and between poly-rising factorial sequences and Lah numbers and the poly-falling factorial sequences. We also derive recurrence formula of these sequences and reciprocal formula of the poly-falling factorial sequences.
Details
- Language :
- English
- ISSN :
- 23915455 and 94334994
- Volume :
- 19
- Issue :
- 1
- Database :
- Directory of Open Access Journals
- Journal :
- Open Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.3d50853c0ee943349942a7629030ef3b
- Document Type :
- article
- Full Text :
- https://doi.org/10.1515/math-2021-0128