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

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.