Gaussian Hypergeometric series and supercongruences

Osburn, Robert; Schneider, Carsten

doi:10.1090/S0025-5718-08-02118-2

Gaussian Hypergeometric series and supercongruences
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by Robert Osburn and Carsten Schneider PDF

Math. Comp. 78 (2009), 275-292 Request permission

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.

References

Scott Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 1999) Dev. Math., vol. 4, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1–12. MR 1880076, DOI 10.1007/978-1-4613-0257-5_{1}
Scott Ahlgren, Shalosh B. Ekhad, Ken Ono, and Doron Zeilberger, A binomial coefficient identity associated to a conjecture of Beukers, Electron. J. Combin. 5 (1998), Research Paper 10, 1 unnumbered page. MR 1600106
Scott Ahlgren and Ken Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187–212. MR 1739404, DOI 10.1515/crll.2000.004
George E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441–484. MR 352557, DOI 10.1137/1016081
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
George E. Andrews, Peter Paule, and Carsten Schneider, Plane partitions. VI. Stembridge’s TSPP theorem, Adv. in Appl. Math. 34 (2005), no. 4, 709–739. MR 2128995, DOI 10.1016/j.aam.2004.07.008
George E. Andrews and K. Uchimura, Identities in combinatorics. IV. Differentiation and harmonic numbers, Utilitas Math. 28 (1985), 265–269. MR 821962
S. Chowla, B. Dwork, and Ronald Evans, On the mod $p^2$ determination of $\left ({(p-1)/2\atop (p-1)/4}\right )$, J. Number Theory 24 (1986), no. 2, 188–196. MR 863654, DOI 10.1016/0022-314X(86)90102-2
Kathy Driver, Helmut Prodinger, Carsten Schneider, and J. A. C. Weideman, Padé approximations to the logarithm. II. Identities, recurrences, and symbolic computation, Ramanujan J. 11 (2006), no. 2, 139–158. MR 2267670, DOI 10.1007/s11139-006-6503-4
Ronald J. Evans, Identities for products of Gauss sums over finite fields, Enseign. Math. (2) 27 (1981), no. 3-4, 197–209 (1982). MR 659148
Sharon Frechette, Ken Ono, and Matthew Papanikolas, Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not. 60 (2004), 3233–3262. MR 2096220, DOI 10.1155/S1073792804132522
M. Fulmek and 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, DOI 10.1006/eujc.1999.0367
J. Fuselier, Hypergeometric functions over finite fields and relations to modular forms and elliptic curves, Ph.D. thesis, Texas A$\&$M University, 2007.
I. M. Gel′fand and M. I. Graev, Hypergeometric functions over finite fields, Dokl. Akad. Nauk 381 (2001), no. 6, 732–737 (Russian). MR 1892519
I. M. Gel′fand, M. I. Graev, and V. S. Retakh, Hypergeometric functions over an arbitrary field, Uspekhi Mat. Nauk 59 (2004), no. 5(359), 29–100 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 5, 831–905. MR 2125927, DOI 10.1070/RM2004v059n05ABEH000771
J. Greene, Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields, Ph.D. thesis, University of Minnesota, 1984.
John Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77–101. MR 879564, DOI 10.1090/S0002-9947-1987-0879564-8
Benedict H. Gross and Neal Koblitz, Gauss sums and the $p$-adic $\Gamma$-function, Ann. of Math. (2) 109 (1979), no. 3, 569–581. MR 534763, DOI 10.2307/1971226
Michael Karr, Summation in finite terms, J. Assoc. Comput. Mach. 28 (1981), no. 2, 305–350. MR 612083, DOI 10.1145/322248.322255
Michael Karr, Theory of summation in finite terms, J. Symbolic Comput. 1 (1985), no. 3, 303–315. MR 849038, DOI 10.1016/S0747-7171(85)80038-9
Timothy Kilbourn, An extension of the Apéry number supercongruence, Acta Arith. 123 (2006), no. 4, 335–348. MR 2262248, DOI 10.4064/aa123-4-3
Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003, DOI 10.1007/978-1-4612-1112-9
Neal Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio Math. 48 (1983), no. 1, 3–23. MR 700577
Masao Koike, Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields, Hiroshima Math. J. 25 (1995), no. 1, 43–52. MR 1322601
Po-Ru Loh and Robert C. Rhoades, $p$-adic and combinatorial properties of modular form coefficients, Int. J. Number Theory 2 (2006), no. 2, 305–328. MR 2240232, DOI 10.1142/S1793042106000590
D. McCarthy, R. Osburn, A $p$-adic analogue of a formula of Ramanujan, submitted.
Eric Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99 (2003), no. 1, 139–147. MR 1957248, DOI 10.1016/S0022-314X(02)00052-5
Eric Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc. 355 (2003), no. 3, 987–1007. MR 1938742, DOI 10.1090/S0002-9947-02-03172-0
Eric 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, DOI 10.1090/S0002-9939-04-07697-X
M. Ram Murty, Introduction to $p$-adic analytic number theory, AMS/IP Studies in Advanced Mathematics, vol. 27, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1913413, DOI 10.1090/amsip/027
Ken Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1205–1223. MR 1407498, DOI 10.1090/S0002-9947-98-01887-X
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
Peter Paule and Carsten Schneider, Computer proofs of a new family of harmonic number identities, Adv. in Appl. Math. 31 (2003), no. 2, 359–378. MR 2001619, DOI 10.1016/S0196-8858(03)00016-2
Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, $A=B$, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk. MR 1379802
Matthew Papanikolas, A formula and a congruence for Ramanujan’s $\tau$-function, Proc. Amer. Math. Soc. 134 (2006), no. 2, 333–341. MR 2175999, DOI 10.1090/S0002-9939-05-08029-9
H. Prodinger, Human proofs of identities by Osburn and Schneider, preprint available at http://front.math.ucdavis.edu/0710.0464.
Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
Carsten 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
Carsten Schneider, Symbolic summation assists combinatorics, Sém. Lothar. Combin. 56 (2006/07), Art. B56b, 36. MR 2317679
C. Schneider, A refined difference field theory for symbolic summation, SFB-Report 2007-24, SFB F013, J. Kepler University Linz, 2007.
John R. Stembridge, The enumeration of totally symmetric plane partitions, Adv. Math. 111 (1995), no. 2, 227–243. MR 1318529, DOI 10.1006/aima.1995.1023
Doron Zeilberger, The method of creative telescoping, J. Symbolic Comput. 11 (1991), no. 3, 195–204. MR 1103727, DOI 10.1016/S0747-7171(08)80044-2
Koichi Yamamoto, On a conjecture of Hasse concerning multiplicative relations of Gaussian sums, J. Combinatorial Theory 1 (1966), 476–489. MR 213311