An application of hypergeometric shift operators to the chi-spherical Fourier transform

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We study the action of hypergeometric shift operators on the Heckman-Opdam hypergeometric functions associated with the BCₙ type root system and some negative multiplicities. Those hypergeometric functions are connected to the χ-spherical functions on Hermitian symmetric spaces U/K where χ is a nontrivial character of K. We apply shift operators to the hypergeometric functions to move negative multiplicities to positive ones. This allows us to use many well-known results of the hypergeometric functions associated with positive multiplicities. In particular, we use this technique to achieve exponential estimates for the χ-spherical functions. The motive comes from the Paley-Wiener type theorem on line bundles over Hermitian symmetric spaces.