Friday, 22 June 2012

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

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