Counting the Number of Solutions to Certain Infinite Diophantine Equations

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