Congruences for 5-regular partitions modulo powers of 5

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 (kge 1) and (nge 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

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