A Reflection Formula for the Gaussian Hypergeometric Function of Matrix Argument

We obtain a reflection formula for the Gaussian hypergeometric function of real symmetric matrix argument. We also show that this result extends to the Gaussian hypergeometric function defined over the symmetric cones, and even to generalizations of the Gaussian hypergeometric function defined in terms of series of Jack polynomials. Finally, we obtain a quadratic transformation formula for the Gaussian hypergeometric function of Hermitian $2 \times 2$ matrix argument.