Asymptotic formulas for general colored partition functions

In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1<s_2<\cdots <s_k$ and ${\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\dots ,{\mathrm{\ell }}_k$. Let $g(\mathbf{s},\mathbf{l},n)$ be the number of $\mathrm{\ell }$-colored partitions of $n$ with ${\mathrm{\ell }}_i$ of the colors appearing only in multiplies of $s_i\ (1\le i\le k)$,where$\ell = \ell_1+\cdots +\ell_k$. By using the elementary method we obtain an asymptotic formula for the partition function $g($\mathbf{s}$, $\mathbf{l}$, n)$ with an explicit error term.