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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Gaussian Hypergeometric series and supercongruences


Authors: Robert Osburn and Carsten Schneider
Journal: Math. Comp. 78 (2009), 275-292
MSC (2000): Primary 11F33, 33F10; Secondary 11S80
Published electronically: April 29, 2008
MathSciNet review: 2448707
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Abstract: 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 $ \mathbb{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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Additional Information

Robert Osburn
Affiliation: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Address at time of publication: IHÉS, Le Bois-Marie, 35, route de Chartres, F-91440 Bures-sur-Yvette, France
Email: robert.osburn@ucd.ie, osburn@ihes.fr

Carsten Schneider
Affiliation: Research Institute for Symbolic Computation, J. Kepler University Linz, Altenberger Strasse 69, A-4040 Linz, Austria
Email: Carsten.Schneider@risc.uni-linz.ac.at

DOI: http://dx.doi.org/10.1090/S0025-5718-08-02118-2
PII: S 0025-5718(08)02118-2
Received by editor(s): April 23, 2007
Received by editor(s) in revised form: November 1, 2007
Published electronically: April 29, 2008
Additional Notes: The second author was supported by the SFB-grant F1305 and the grant P16613-N12 of the Austrian FWF
Article copyright: © Copyright 2008 American Mathematical Society