Hypergeometric function of a matrix argument

Definition

Let $m \ge 1$ be an integer, let $\alpha > 0$, and let $X$ be a complex $m \times m$ symmetric matrix with eigenvalues $x_1, \dots, x_m$. Write $\kappa \vdash k$ when $\kappa = (\kappa_1 \ge \kappa_2 \ge \dots)$ is an integer partition of $k$. Let $C_\kappa^{(\alpha)}(X)$ be the Jack polynomial of $\kappa$ in $x_1, \dots, x_m$, normalised so that $\sum_{\kappa \vdash k} C_\kappa^{(\alpha)}(X) = (\operatorname{tr} X)^k$. It is $0$ when $\kappa$ has more than $m$ nonzero parts. For a complex number $a$, the generalized Pochhammer symbol is $$(a)_\kappa^{(\alpha)} = \prod_{i \ge 1} \prod_{j = 1}^{\kappa_i} \left( a - \frac{i - 1}{\alpha} + j - 1 \right).$$ Let $a_1, \dots, a_p$ and $b_1, \dots, b_q$ be complex numbers such that $(b_l)_\kappa^{(\alpha)} \ne 0$ for every $l$ and every $\kappa$ with at most $m$ parts. The hypergeometric function of matrix argument is the series $${}_pF_q^{(\alpha)}(a_1, \dots, a_p; b_1, \dots, b_q; X) = \sum_{k = 0}^{\infty} \sum_{\kappa \vdash k} \frac{(a_1)_\kappa^{(\alpha)} \cdots (a_p)_\kappa^{(\alpha)}}{(b_1)_\kappa^{(\alpha)} \cdots (b_q)_\kappa^{(\alpha)}} \frac{C_\kappa^{(\alpha)}(X)}{k!},$$ wherever it converges.