A Look at the Reciprocal of the Generating Function for $p$-Regular Partitions

Abstract

The generating function for $p$-regular partitions is given by $\frac{{(q^p;q^p)}_{\mathrm{\infty }}}{{(q;q)}_{\mathrm{\infty }}}$ .In this paper, we will investigate the reciprocal of this generating function. Several interesting results will be presented, and as a corollary of one of these, we will get a parity result due to Sellers for $p$-regular partitions with distinct parts.