Characterizing the number of coloured $m$-ary partitions modulo $m$, with and without gaps

Abstract

In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization for the number of $m$-ary partitions modulo $m$, with and without gaps. In this paper we extend these results to the case of coloured $m$-ary partitions, with and without gaps. Our method of proof is different, giving explicit expansions for the generating functions modulo $m$