$n$.

The proof is similar to that of [@O3 Proposition (9.4)(iii)], and follows from a counting argument using Theorem \[stcore\]. Each $(s,t)$-core partition ${\lambda}$ of $n$ is completely and uniquely determined by its $g$-core ${\mathcal Cor}_g({\lambda})$ and $g$-quotient ${\mathcal Quo}_g({\lambda})$. And, by Theorem \[stcore\], if ${\lambda}$ has $g$-weight $w$, then ${\mathcal Quo}_g({\lambda})$ can be any $g$-tuple of $(s',t')$-cores of weight $w$. We have $$\label{jade}
\hat{\psi}_{s,t}(n)={\displaystyle}\sum_{w\geq 0} Q_{s',t'}(g,w)f_g(n-gw).$$ Equation (\[jade\]) can be transformed into the generating function $$\psi_{s,t}(x)= \psi_{s',t'}(x^g)^g  F_g(x),$$ as claimed.

Simultaneous self-conjugate cores with common divisor.
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We consider self-conjugate $(s,t)$-core partitions, or $(s,t)^*$[*-core*]{} partitions. Our description of the core and quotient of a self-conjugate $(s,t)$-core partition is a consequence of the following lemma which appears in [@HN].

\[nec0\] Let $t$ be any positive integer, $\lambda$ be a partition, and ${\mathcal Quo}_t({\lambda})=({\lambda}_0, \ldots , {\lambda}_{t-1})$. Then $\lambda$ is self-conjugate if and only if ${\mathcal Cor}_t({\lambda})$ is self-conjugate and $\lambda^{\vee}_{i}=\lambda_{t-i-1}$ for all $0 \leq i \leq t-1$.

\[nec\] A partition $\lambda$ is an self-conjugate $(s,t)$-core partition if and only if ${\mathcal Cor}_g(\lambda)$ is self-conjugate and each part of ${\mathcal Quo}_g(\lambda)=(\lambda_0,\cdots,\lambda_{g-1})$ is an $(s',t')$-core such that $\lambda_i=\lambda^{\vee}_{g-i-1}$.

This follows from Theorem \[stcore\] and Lemma \[nec0\].

Let $\hat{\psi}^*_{s,t}(n)$ be the number of 