Congruences for $(2, 5)$-regular Bipartitions into Distinct Parts

Abstract

Let $B_{2, 5}(n)$ denote the number of $(2, 5)$-regular bipartitions of a positive integer $n$ into distinct parts . In this paper, we establish several infinite families of congruences modulo powers of $2$ for $B_{2, 5}(n)$. For example,
\begin{equation*}
\sum_{n=0}^{\infty}B_{2, 5}\lb(2^{2\alpha+1}\cdot5^{2\beta}n+\frac{2^{2\alpha+2}\cdot5^{2\beta}-1}{3}\rb)q^n \equiv 2f_1f_{5}^3 \pmod{2^2}, \end{equation*}
for $\alpha, \beta \geq 0$.