Asymptotic formulas for general colored partition functions

Abstract

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 ${\ell }_1,{\ell }_2,\dots ,{\ell }_k$. 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.

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