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 ${}_{1}ψ_{1}$ formula and Bailey's ${}_{6}ψ_{6}$ 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 $B_{3}(n)$.

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