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 $1=s_1<s_2<\dots <s_k$ and $\ell_1, \ell_2,\dots , \ell_k$. Let $g(\mathbf{s}, \mathbf{l}, n)$ be the number of $\ell$-colored partitions of $n$ with $\ell_i$ 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.