Congruences for the Coefficients of the Powers of the Euler Product

Abstract

Let $p_k(n)$ be given by the $k$-th power of the Euler Product ${\prod }_{n=1}^{\infty }(1-q^n{)}^k={\sum }_{n=0}^{\infty }{p}_k(n)q^n$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\alpha }n+\frac{k(2^{2\alpha }-1)}{3})$ $(1\le k\le 3)$ and $p_{3k}(3^{2\beta }n+\frac{k(3^{2\beta }-1)}{8})$ $(1\le k\le 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite families of congruences for $p_k(n)$ modulo any $m\ge 2$, where $1\le k\le 24$ and $3|k$ or $8|k$. Based on these congruences for $p_k(n)$, infinite families of congruences for many partition functions such as the overpartition function, $t$-core partition functions and $\ell$-regular partition functions are easily obtained.

26 pages, replaced references, corrected typos