Gaussian hypergeometric series and supercongruences

Details Gaussian hypergeometric series and supercongruences Gaussian hypergeometric series and supercongruences

2006-10-09

arxiv.org/abs/math/0610281v2

Mathematics of Computation 78 (2009), 275-292

Mathematics

Number Theory

completely rewritten version, 19 pages, accepted for publication in Mathematics of Computation

Scientific paper

Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of F_p points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to supercongruences. This application uses two non-trivial generalized Harmonic sum identities discovered using the computer summation package Sigma. We illustrate the usage of Sigma in the discovery and proof of these two identities.

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