Congruences for 5-regular partitions modulo powers of 5

Abstract

Let $b_{5}(n)$ denote the number of 5-regular partitions of n. We find the generating functions of $b_{5}(An+B)$ for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for $b_{5}(n)$ modulo powers of 5. For example, for any integers $k\ge 1$ and $n\ge 0$, we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$