Some new congruences for \((j, k)\)-regular bipartition triples
A partition of a positive integer \(n\) is a non-increasing sequence of positive integers called parts whose sum is \(n\). Denote by \(p(n)\) the number of all partitions of a positive integer \(n\). Motivated by Ramanujan's congruences on \(p(n)\), the arithmetic propertities of partition with certain restriction have received much attention, such as \(\ell\)-regular partition of a positive integer \(n\). For a positive integer \(\ell\), a partition of \(n\) is called \(\ell\)-regular if none of its parts is divisible by \(\ell\). A partition triple \((\zeta_1,\zeta_2,\zeta_3)\) of a positive integer is called an \(\ell\)-regular partition triple if each \(\zeta_i \), \((i=1,2,3)\) is an \(\ell\)-regular partition. A bipartition triple \((\kappa,\xi)\) of \(n\) is called \((j,k)\)-regular if \(\kappa\) is \(j\)-regular and \(\xi\) is \(k\)-regular. Let \(BT_{j,k}(n)\) denote the number of \((j,k)\)-regular bipartition triples of a positive integer \(n\). Clearly, the generating function for \(BT_{j,k}(n)\) is given by \[\sum\limits_{n=0}^{\infty}BT_{j,k}(n)q^n=\frac{(q^j;q^j)_{\infty}^3(q^k;q^k)_{\infty}^3}{(q;q)_{\infty}^6}.\] Recently, \textit{M. S. Mahadeva Naika} and \textit{S. S. Nayaka} [Afr. Mat. 31, No.2, 249--261 (2020; Zbl 1449.05023)] deduced some congruences modulo \(3,9\) and \(27\) for \(BT_{j,k}(n)\) for \((j,k)\in\{(2,3), (2,6),(3,3),(3,5),(3,7),(3,9),(9,9)\}\). In the paper under review, via generating function manipulations and \(q\)-series identities, the authors prove some infinite families of congruences modulo \(3,4,5,7,8\) and \(9\) for \(BT_{j,k}(n)\) with \((j,k)\in\{(2,4), (3,4),(3,5),(3,6),(3,7),(3,11),(5,5\alpha),(5,27),(7,7)\}\), which extend the list of proven arithmetic properties satisfied by \(BT_{j,k}(n)\).
scientific article; zbMATH DE number 7608543
scientific article