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Hypergeometric type identities in the \(p\)-adic setting and modular forms. (English) Zbl 1397.11078

Summary: We prove hypergeometric type identities for a function defined in terms of quotients of the \( p\)-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated \( _4F_3\) hypergeometric series and the Fourier coefficients of a certain weight four modular form.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
33C20 Generalized hypergeometric series, \({}_pF_q\)
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)

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