$M$-ary partition polynomials

Abstract

Let $M=(m_{i})_{i=0}^{\infty}$ be a sequence of integers such that $m_{0}=1$ and $m_{i}\geq 2$ for $i\geq 1$. In this paper we study $M$-ary partition polynomials $(p_{M}(n,t))_{n=0}^{\infty}$ defined as the coefficient in the following power series expansion: \begin{align*} \prod_{i=0}^{\infty}\frac{1}{1-tq^{M_{i}}} = \sum_{n=0}^{\infty} p_{M}(n,t)q^{n}, \end{align*} where $M_{i}=\prod_{j=0}^{i}m_{j}$. In particular, we provide a detailed description of their rational roots and show, that all their complex roots have absolute values not greater than $2$. We also study arithmetic properties of $M$-ary partition polynomials. One of our main results says that if $n=a_{0}+a_{1}M_{1}+\cdots +a_{k}M_{k}$ is a (unique) representation such that $a_{j}\in\{0,\ldots ,m_{j+1}-1\}$ for every $j$, then \begin{align*} p_{M}(n,t)\equiv t^{a_{0}}\prod t^{a_{j}}f(a_{j}+1,t^{m_{j}-1}) \pmod{g_{k}(t)}, \end{align*} where $f(a,t):=\frac{t^{a}-1}{t-1}$ and $g_{k}(t):=\gcd \big(t^{m_{1}+m_{2}-1}f(m_{2},t^{m_{1}-1}),\ldots ,t^{m_{k}+m_{k+1}-1}f(m_{k+1},t^{m_{k}-1})\big)$. This is a polynomial generalisation of the well-known characterisation modulo $m$ of the sequence of $m$-ary partition.