## 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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### References:

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