Definition of Hypergeometric Function of Matrix Argument

$$_pF^q(a_1 \cdots a_p;b_1 \cdots b_q;X)=\sum_{k=0}^{\infty}\sum_{\kappa} \frac{(a_1)_{\kappa} \cdots (a_p)_{\kappa}}{(b_1)_{\kappa} \cdots (b_q)_{\kappa}}\,\frac{C_{\kappa}(X)}{k!}$$
where the Generalized Hypergeometric Coefficient is Given by

$$(a)_{\kappa}=\prod _{i=1}^m \left(a-\frac{1}{2}(i-1){} \right )_{k_i}$$ , Where the Pochammer Symbol

$$(\alpha)_j=\alpha (\alpha+1) \cdots (\alpha+j-1),\;(\alpha)_0=1$$

Definition of Zonal Polynomial

Let $Y$ be $m \times m$ Symmetric Matrix with Latent Roots(Eigen values) $y_1 \cdots y_m$.

Let $$\kappa=\left(k_1 \cdots k_m \right)$$

be the Partition of an Integer $k$ with Partition Size not more than $m$.

The Zonal Polynomial Corresponding to $\kappa$ Denoted by $C_{\kappa}(Y)$ is a Symmetric , Homogeneous Polynomial of Degree $k$ in Latent Roots $y_1 \cdots y_m$ Such that:

$$C_{\kappa}(Y)= d_k\,y_1^{k_1} \cdots y_m^{k_m}+ \text{Terms of Lower Weight}$$ , $d_k$ being a constant

$C_{\kappa}(Y)$ is an Eigen Function of Differential Operator $\Delta_{Y}$ given by:

$$\Delta_Y=\sum_{i=1}^{m}y_i^2 \frac{\partial^2 }{\partial y_i^2}+\sum_{i=1}^m \sum_{j=1,j \neq i}^m \frac{y_i^2}{y_i-y_j}\frac{\partial }{\partial y_i}$$

As $\kappa$ varies over all Partitions of $k$,The Zonal Polynomials have Unit Coefficients in the Expansion of $Tr(Y)^k$ i.e.,

$$Tr(Y)^k= \left(y_1+ \cdots +y_m \right)^k= \sum_{\kappa} C_{\kappa}(Y)$$