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We define a function which extends Gaussian hypergeometric series to the$p$-adic setting. This new function allows results involving Gaussianhypergeometric series to be extended to a wider class of primes. We demonstratethis by providing various congruences between the function and truncatedclassical hypergeometric series. These congruences provide a framework forproving the supercongruence conjectures of Rodriguez-Villegas.

Extending Gaussian hypergeometric series to the $p$-adic setting