Congruences modulo powers of 5 for two restricted bipartitions

Let \(B_{5}(n)\) denote the number of 5-regular bipartitions of n. We establish some Ramanujan-type congruences like \(B_{5}(4n+3) \equiv 0\) (mod 5) and many infinite families of congruences for \(B_{5}(n)\) modulo higher powers of 5 such as

$$\begin{aligned} B_{5}\left( 5^{2k-1}n+\frac{2\cdot 5^{2k-1}-1}{3}\right) \equiv 0 \pmod {5^k}. \end{aligned}$$

We also apply the same method to obtain some similar results for another type of bipartition function. Meanwhile, we give a new interesting interlinked q-series identity related with Rogers–Ramanujan continued fraction, which answers a question of M. Hirschhorn.