integral involving hypergeometric function of matrix argument

Question

This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to this observation, but I open a new question because yet I'm not sure it is exactly the same).

Let $p \geq 1$ be an integer, $a,b > \frac{p-1}{2}$, $\Theta$ a positive scalar $p \times p$-matrix $\Theta = \text{diag}(\theta, \ldots, \theta)$ and $U$ a symmetric $p \times p$-matrix satisfying $0 < U < I_p$. The conjecture is: $$ \int_{S >0} {\det(S)}^{a+b-\frac12(p+1)} \exp\left(-\mathrm{tr}\left(\frac{S}{2}\right)\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta S^\frac12 U S^\frac12\right)\textrm{d}S \\ = 2^{a+b}\Gamma_p(a+b){}_1\!F_1(a+b, b, \Theta U). $$ According to this paper by Constantine (page 1280), the integral in the LHS is difficult to evaluate unless $\Theta$ is a scalar matrix. This suggests that this integral can be simplified but I don't find the result in the literature.

Do you have a reference for this result, or a proof?

Answer 1

The zonal polynomial $C_\kappa$ satisfies $C_\kappa(SR) = C_\kappa(S^\frac12 R S^\frac12)$, for $S$ symmetric positive definite and $R$ symmetric, according to (1.5.3) in Gupta & Nagar's book. This implies that the same relation holds for the hypergeometric functions, in view of their definition in terms of the zonal polynomials. Thus, since $\Theta$ is scalar, $$ {}_0\!F_1\left(b, \frac{1}{2}\Theta S^\frac12 U S^\frac12\right) = {}_0\!F_1\left(b, \frac{1}{2}\Theta S U\right). $$ Now, it is known that $$ \int_{S>0}\exp\bigl(\text{tr}(S)\bigr)\det(S)^{\alpha-\frac12(p+1)}{}_0\!F_1(b,ST)\text{d}S \\ = \Gamma_p(\alpha){}_1\!F_1(\alpha,b,T) $$ (particular case of the known relation between ${}_m\!F_n$ and ${}_{m+1}\!F_n$, theorem 1.6.2 in Gupta & Nagar's book).

The result follows from these two relations, after the change of variables $S'=S/2$ in the integral of the OP.