Explicit Formulas for Partition Pairs and Triples with 3-Cores

Let $A_3(n)$ (resp. $B_3(n)$) denote the number of partition pairs (resp. triples) of $n$ where each partition is 3-core. By applying Ramanujan’s ${}_1{\psi }_1$ formula and Bailey’s ${}_6{\psi }_6$ formula, we find the explicit formulas for $A_3(n)$ and $B_3(n)$. Using these formulas, we confirm a conjecture of Xia and establish many arithmetic identities satisfied by $A_3(n)$ and $B_3(n)$.