On $p$-adic valuations of certain $m$ colored $p$-ary partition functions

Let $k\in {\mathbb{N}}_{\ge 2}$ and for given $m\in \mathbb{Z}\setminus \{0\}$ consider the sequence $(S_{k,m}(n){)}_{n\in \mathbb{N}}$ defined by the power series expansion $$\frac{1}{(1-x{)}^m}\prod _{i=0}^{\mathrm{\infty }}\frac{1}{(1-x^{k^i}{)}^m}=\sum _{n=0}^{\mathrm{\infty }}{S}_{k,m}(n)x^n\mathrm{.}$$ The number $S_{k,m}(n)$ for $m\in {\mathbb{N}}_{+}$ has a natural combinatorial interpretation: it counts the number of representations of $n$ as sums of powers of $k$, where the part equal to $1$ takes one among $mk$ colors and each part $>1$ takes $m(k-1)$ colors. We concentrate on the case when $k=p\in \mathbb{P}$. Our main result is the computation of the exact value of the $p$-adic valuation of $S_{p,m}(n)$. In particular, in each case the set of values of ${\nu }_p(S_{p,m}(n))$ is finite and the maximum value is bounded by $\text{op}max\{{\nu }_p(m)+1,{\nu }_p(m+1)+1\}$. Our results can be seen as a generalization of earlier work of Churchhouse and recent work of Gawron, Miska and Ulas, and the present authors.