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Extending Gaussian hypergeometric series to the \(p\)-adic setting. (English) Zbl 1253.33024

The author introduces a \(p\)-adic extension of the Gaussian hypergeometric series. The main results of the paper are four families of congruences \(\pmod{p^2}\) and \(\pmod{p^3}\), between certain \(p\)-adic hypergeometric series and truncated classical generalized hypergeometric series, where \(p\) is an odd prime number. Among the tools for the proofs of these results are several properties of the \(p\)-adic gamma function and its logarithmic derivatives.

MSC:

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