On the restricted $k$-multipartition function

Abstract

Let $\mathbf a=(a_1,\ldots,a_r)$ be a sequence of positive integers and $k\geq 2$ an integer. We study $p_{k,\mathbf a}(n)$, the restricted $k$-multipartition function associated to $\mathbf a$ and $k$. We prove new formulas for $p_{k,\mathbf a}(n)$, its waves $W_j(n,k,\mathbf a)$'s and its polynomial part $P_{k,\mathbf a}(n)$. Also, we give a lower bound for the density of the set $\{n\geq 0\;:\;p_{k,\mathbf a}(n)\not\equiv 0(\bmod\;m)\}$, where $m\geq 2$ is an integer.

Theorems & Definitions (25)

• Proposition 2.1
• proof
• Remark 2.2
• Lemma 2.3
• Proposition 2.4
• Proposition 2.5
• proof
• Corollary 2.6
• proof
• Proposition 2.7