Arithmetic of the 7-regular bipartition function modulo 3

In this paper, we study the function Bℓ(n) which counts the number of ℓ-regular bipartitions of n. Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). In particular, using Ramanujan’s two modular equations of degree 7, we prove an infinite family of congruences: for α≥2 and n≥0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_7 \biggl(3^{\alpha}n+\frac{5\cdot 3^{\alpha-1}-1}{2} \biggr)\equiv 0\ ({ \rm mod\ }3). $$\end{document} In addition, we give an elementary proof of two infinite families of congruences modulo 3 satisfied by the 7-regular partition function due to Furcy and Penniston (Ramanujan J. 27:101–108, 2012). We also present two conjectures for B13(n) modulo 3.