ARITHMETIC PROPERTIES OF -REGULAR BIPARTITIONS

Abstract

Let$B_{k,\ell }(n)$denote the number of$(k,\ell )$-regular bipartitions of$n$. Employing both the theory of modular forms and some elementary methods, we systematically study the arithmetic properties of$B_{3,\ell }(n)$and$B_{5,\ell }(n)$. In particular, we confirm all the conjectures proposed by Dou [‘Congruences for (3,11)-regular bipartitions modulo 11’,Ramanujan J.40, 535–540].