Hook lengths in self-conjugate partitions

In 2010, G.-N. Han obtained the generating function for the number of size $t$ hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even $t$. If $n_t(\lambda )$ is the number of size $t$ hooks in a partition $\lambda ,$ then for even $t$ we have $$\sum _{\lambda \in \mathcal{S}\mathcal{C}}{x}^{n_t(\lambda )}{q}^{\mathrm{\mid }\lambda \mathrm{\mid }}=(-q;q^2{)}_{\mathrm{\infty }}\cdot ((1-x^2)q^{2t};q^{2t}{)}_{\mathrm{\infty }}^{\frac{t}{2}}\mathrm{.}$$ As a consequence, if $a_t^{\ast }(n)$ is the number of such hooks among the self-conjugate partitions of $n,$ then for even $t$ we obtain the simple formula $$a_t^{\ast }(n)=t\sum _{j\ge 1}{q}^{\ast }(n-2tj),$$ where $q^{\ast }(m)$ is the number of partitions of $m$ into distinct odd parts. As a corollary, we find that $t\mid a_t^{\ast }(n),$ which confirms a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.