Congruences for the coefficients of the powers of the Euler Product

Abstract

Let $p_k(n)$ be given by the series expansion of 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.

Appendix

See Tables 3, 4 and 5.

Table 3 The values of f(k), g(k) and the initial values of $A_k(\alpha )$ and $B_k(\alpha )$

Table 4 The values of h(k), r(k) and the initial values of $C_k(\beta )$ and $D_{k,i}(\beta )$

Table 5 Values for $\mu _m(k)$ and $\nu _m(k)$