Congruence properties modulo powers of 2 for 4-regular partitions

Let b(& ell;)(n) denote the number of -regular pound partitions of n. Congruences properties modulo powers of 2 for b4(n) have been considered subsequently by Andrews- Hirschhorn-Sellers, Chen, Cui-Gu, Xia, Dai, and Ballantine-Merca. In this paper, we present an approach which can be utilized to prove "self-similar" congruence property satisfied by the generating function of b(4)(n). As an immediate consequence, one can obtain dozens of congruence families modulo powers of 2 enjoyed by b(4)(n). These results not only generalize some previous results, but also can be viewed as a supplement to Keith and Zanello's comprehensive study of the congruence properties for & ell;-regular partition functions. Finally, we also pose several conjectures on congruence families, internal congruence families and self-similar congruence properties for 4-, 8- and 16-regular partition functions.