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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
DOI: https://doi.org/10.1090/S0025-5718-08-02118-2
Published electronically: April 29, 2008
MathSciNet review: 2448707
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Abstract | References | Similar Articles | Additional Information

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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  • 1. S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, Fl, 1999), 1-12, Dev. Math., 4, Kluwer, Dordrecht, 2001. MR 1880076 (2003i:33025)
  • 2. S. Ahlgren, S. Ekhad, K. Ono, D. Zeilberger, A binomial coefficient identity associated to a conjecture of Beukers, Electron. J. Combin. 5 (1998), Research Paper 10, 1 p. MR 1600106
  • 3. S. Ahlgren, K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187-212. MR 1739404 (2001c:11057)
  • 4. G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441-484. MR 0352557 (50:5044)
  • 5. G. Andrews, R. Askey, R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
  • 6. G. Andrews, P. Paule, C. Schneider, Plane partitions. VI. Stembridge's TSPP theorem, Adv. in Appl. Math. 34 (2005), no. 4, 709-739. MR 2128995 (2006b:05012)
  • 7. G. E. Andrews and K. Uchimura, Identities in combinatorics. IV. Differentiation and harmonic numbers, Utilitas Math. 28 (1985), 265-269. MR 821962 (87e:05014)
  • 8. S. Chowla, B. Dwork, and R. Evans, On the mod $ p^2$ determination of $ \binom{(p-1)/2}{(p-1)/4}$, J. Number Th. 24 (1986), no. 2, 188-196. MR 863654 (88a:11130)
  • 9. K. Driver, H. Prodinger, C. Schneider, J. Weideman, Padé approximations to the logarithm. II. Identities, recurrences, and symbolic computation, Ramanujan J. 11 (2006), no. 2, 139-158. MR 2267670 (2007i:05017)
  • 10. R. Evans, Identities for products of Gauss sums over finite fields, Enseign. Math. 27 (1981), 197-209. MR 659148 (83i:10050)
  • 11. S. Frechette, K. Ono, and M. Papanikolas, Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not. 2004, no. 60, 3233-3262. MR 2096220 (2006a:11055)
  • 12. M. Fulmek, C. Krattenthaler, The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis. II, European J. Combin. 21 (2000), no. 5, 601-640. MR 1771987 (2002e:05039)
  • 13. J. Fuselier, Hypergeometric functions over finite fields and relations to modular forms and elliptic curves, Ph.D. thesis, Texas A$ \&$M University, 2007.
  • 14. I. Gel'fand, M. Graev, Hypergeometric functions over finite field, Dokl. Akad. Nauk 381 (2001), no. 6, 732-737. MR 1892519 (2003f:33020)
  • 15. I. Gel'fand, M. Graev, and V. Retakh, Hypergeometric functions over an arbitrary field, (Russian) Uspekhi Mat. Nauk 59 (2004), no. 5 (359), 29-100; translation in Russian Math. Surveys 59 (2004), no. 5, 831-905. MR 2125927 (2006e:33022)
  • 16. J. Greene, Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields, Ph.D. thesis, University of Minnesota, 1984.
  • 17. J. Greene, Hypergeometric series over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77-101. MR 879564 (88e:11122)
  • 18. B. Gross, N. Koblitz, Gauss sums and the $ p$-adic $ \Gamma$-function, Ann. of Math. (2) 109 (1979), no. 3, 569-581. MR 534763 (80g:12015)
  • 19. M. Karr, Summation in finite terms, J. Assoc. Comput. Mach. 28 (1981), no. 2, 305-350. MR 612083 (82m:12018)
  • 20. M. Karr, Theory of summation in finite terms, J. Symbolic Comput. 1 (1985), no. 3, 303-315. MR 849038 (89a:12016)
  • 21. T. Kilbourn, An extension of the Apéry number supercongruence, Acta Arith. 123 (2006), 335-348. MR 2262248 (2007e:11049)
  • 22. N. Koblitz, $ p$-adic numbers, $ p$-adic analysis, and zeta functions, Second Edition. Graduate Texts in Mathematics 58, Springer-Verlag, New York, 1984. MR 754003 (86c:11086)
  • 23. N. Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio Math. 48 (1983), 3-23. MR 700577 (85d:14034)
  • 24. M. Koike, Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields, Hiroshima Math. J. 25 (1995), 43-52. MR 1322601 (96b:11079)
  • 25. P. Loh, R. Rhoades, $ p$-adic and combinatorial properties of modular form coefficients, Int. J. Number Theory 2 (2006), no. 2, 305-328. MR 2240232 (2007f:11047)
  • 26. D. McCarthy, R. Osburn, A $ p$-adic analogue of a formula of Ramanujan, submitted.
  • 27. E. Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Th. 99 (2003), no. 1, 139-147. MR 1957248 (2004e:11089)
  • 28. E. Mortenson, Supercongruences between truncated $ {}_{2}F_{1}$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc. 355 (2003), 987-1007. MR 1938742 (2003i:11119)
  • 29. E. Mortenson, Supercongruences for truncated $ {}_{n+1}F_{n}$ hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc. 133 (2005), no. 2, 321-330. MR 2093051 (2005f:11080)
  • 30. R. Murty, Introduction to p-adic analytic number theory, AMSCIP Studies in Advanced Mathematics, vol. 27, American Mathematical Society, Providence, RI, 2002. MR 1913413 (2003c:11151)
  • 31. K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), 1205-1223. MR 1407498 (98e:11141)
  • 32. K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and $ q$-series, Amer. Math. Soc., CBMS Regional Conf. in Math., vol. 102, 2004. MR 2020489 (2005c:11053)
  • 33. P. Paule, C. Schneider, Computer proofs of a new family of harmonic number identities, Adv. in Appl. Math. 31 (2003), no. 2, 359-378. MR 2001619 (2004f:33043)
  • 34. M. Petkovšek, H. S. Wilf, and D. Zeilberger, $ A=B$, A. K. Peters, Wellesley, MA, 1996. MR 1379802 (97j:05001)
  • 35. M. Papanikolas, A formula and a congruence for Ramanujan's $ \tau$-function, Proc. Amer. Math. Soc. 134 (2006), no. 2, 333-341. MR 2175999 (2007d:11046)
  • 36. H. Prodinger, Human proofs of identities by Osburn and Schneider, preprint available at http://front.math.ucdavis.edu/0710.0464.
  • 37. A. Robert, A course in $ p$-adic analysis, Graduate Texts in Mathematics 198, Springer-Verlag, New York, 2000. MR 1760253 (2001g:11182)
  • 38. C. Schneider, The summation package Sigma: Underlying principles and a rhombus tiling application, Discrete Math. Theor. Comput. Sci., 6 (2004), no. 2, 365-386. MR 2081481 (2005e:68270)
  • 39. C. Schneider, Symbolic Summation Assists Combinatorics, Sem. Lothar. Combin., 56:1-36, 2007.
    Article B56b. MR 2317679
  • 40. C. Schneider, A refined difference field theory for symbolic summation, SFB-Report 2007-24, SFB F013, J. Kepler University Linz, 2007.
  • 41. J. Stembridge, The enumeration of totally symmetric plane partitions, Adv. Math. 111 (1995), no. 2, 227-243. MR 1318529 (96b:05010)
  • 42. D. Zeilberger, The method of creative telescoping, J. Symbolic Comput. 11 (1991), no. 3, 195-204. MR 1103727 (92c:33005)
  • 43. K. Yamamoto, On a conjecture of Hasse concerning multiplicative relations of Gaussian sums, J. Combin. Theory Ser. A 1 (1966), 476-489. MR 0213311 (35:4175)

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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: https://doi.org/10.1090/S0025-5718-08-02118-2
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

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