Recurrence sequences connected with the $m$--ary partition function and their divisibility properties

Abstract

In this paper we introduce a class of sequences connected with the $m$--ary partition function and investigate their congruence properties. In particular, we get facts about the sequences of $m$--ary partitions $(b_{m}(n))_{m\in\mathbb{N}}$ and $m$--ary partitions with no gaps $(c_{m}(n))_{m\in\mathbb{N}}$. We prove, for example, that for any natural number $2<h\leq m+1$ in both sequences $(b_{m}(n))_{m\in\mathbb{N}}$ and $(c_{m}(n))_{m\in\mathbb{N}}$ any residue class modulo $h$ appears infinitely many times. Moreover, we give new proofs of characterisations modulo $m$ in terms of base--$m$ representation of $n$ for sequences $(b_{m}(n))_{m\in\mathbb{N}}$ and $(c_{m}(n))_{m\in\mathbb{N}}$. We also present a general method of finding such characterisations modulo any power of $m$. Using our approach we get description of $(b_{m}(n)\mod{\mu_{2}})_{n\in\mathbb{N}}$, where $\mu_{2}=m^{2}$ if $m$ is odd and $\mu_{2}=m^{2}/2$ if $m$ is even.