Arithmetic properties of 7-regular partitions

Let (b_{ell }(n)) denote the number of (ell )-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for (b_{7}(n)) modulo powers of 7: for (nge 0) and (jge 1),

$$begin{aligned} b_{7}left( 7^{2j-1}n+frac{3cdot 7^{2j}-1}{4}right) equiv 0 pmod {7^j}. end{aligned}$$

We also find some infinite families of congruences modulo 2 and 7 satisfied by (b_{7}(n)).