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

A Reflection Formula for the Gaussian Hypergeometric Function of Matrix Argument