CONNECTION BETWEEN PARTIAL BELL POLYNOMIALS AND \boldmath$(q ; q)_{k}$; PARTITION FUNCTION, AND CERTAIN \boldmath$ q $-HYPERGEOMETRIC SERIES

Print ISSN: 2319-1023 | Online ISSN: 2582-5461 | Total Downloads : 67

Volume : 10 , Issue : 1
DOI: https://doi.org/10.56827/JRSMMS.2022.1001.1
Author :
M. A. Pathan (Centre for Mathematical and Statistical Sciences, Peechi Campus, Peechi - 680653, Kerala, INDIA)
J. D. Bulnes (Departamento de Ciencias Exatas e Tecnologia, Universidade Federal do Amapa, Rod. Juscelino Kubitschek, Jardin Marco Zero, 68903-419, Macapa, AP, BRAS)
J. L'opez-Bonilla (ESIME-Zacatenco, Instituto Politecnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista 07738 CDMX, MEXICO)
Hemant Kumar (Department of Mathematics, D. A-V. Postgraduate College, Kanpur - 208001, (U.P.), INDIA)

Abstract

We exhibit a relationship between $q$-shifted factorial, $(q ; q)_{n}$, and the incomplete exponential Bell polynomials and also evaluate several $q$-hypergeometric series using the $q$-version of Petkovsek-WilfZeilberger's algorithm. Finally, we write the partition function $p(n)$ in terms of $Q_{m}(k)$, the number of partitions of $m$ using (possibly repeated) parts that do not exceed $k$.

Keywords and Phrases

Partial Bell polynomials, $q$-analysis, Hessenberg determinant, $q$-Hypergeometric series, $q$-Petkovsek-Wilf-Zeilberger's techniques, Partition functions.

A.M.S. subject classification

33D90, 33D70.

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