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On the adelic Gaussian hypergeometric function
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We define the adelic hypergeometric function of special Gaussian type by means of a tower of hypergeometric curves. This function takes values in an adelic completed group ring and interpolates all the hypergeometric functions of the same type over all finite fields. It specializes at the unit argument to the adelic beta function of Ihara and Anderson. We prove some transformation formulas and a summation formula for the adelic hypergeometric function, which are known classically for complex hypergeometric functions.
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Transformation formula of Dwork's $p$-adic hypergeometric function
For a in N^{-1}Z with p not dividing N and 0<a<1, Dwork's p-adic hypergeometric function satisfies F(t)=((-1)^(d+1)t)^l F(t^{-1}) for all d, with a precise sign rule when p=2.
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