CONNECTION BETWEEN PARTIAL BELL POLYNOMIALS AND (q; q)k; PARTITION FUNCTION, AND CERTAIN q-HYPERGEOMETRIC SERIES

M. A. . Pathan; J. D. . Bulnes; J. . L´opez-Bonilla; Hemant . Kumar

doi:10.56827/JRSMMS.2022.1001.1

M. A. . Pathan Centre for Mathematical and Statistical Sciences, Peechi Campus, Peechi - 680653, Kerala, INDIA
J. D. . Bulnes Departamento de Ciencias Exatas e Tecnolog´ıa, Universidade Federal do Amap´a, Rod. Juscelino Kubitschek, Jardin Marco Zero, 68903-419, Macap´a, AP, BRASIL
J. . L´opez-Bonilla ESIME-Zacatenco, Instituto Polit´ecnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista 07738 CDMX, M´EXICO
Hemant . Kumar Department of Mathematics, D. A-V. Postgraduate College, Kanpur - 208001, (U.P.), INDIA

DOI: https://doi.org/10.56827/JRSMMS.2022.1001.1

Keywords: Partial Bell polynomials, q-analysis, Hessenberg determinant, q-Hypergeometric series, q-Petkovsek-Wilf-Zeilberger’s techniques, Partition functions.

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 Qm(k), the number of partitions of m using (possibly repeated) parts that do not exceed k.