  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 $B_{3}(n)$.
