Counting the number of solutions to certain infinite Diophantine equations

Abstract

Let $r, v, n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations $$ n=1^{r}\cdot |k_{1}|^{v}+2^{r}\cdot |k_{2}|^{v}+3^{r}\cdot |k_{3}|^{v}+\ldots, $$ for ${\bf k}=(k_1,k_2,k_3,\dots)\in\mathbb{Z}^{\infty}$. For each $(r,v)\in\mathbb{N}\times\{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.