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A \(p\)-adic supercongruence conjecture of van Hamme. (English) Zbl 1171.11061

The author proves the following conjecture of von Hamme: Let \(p\) be an odd prime; then \[ \sum^{(p-1)/2}_{k=0} (4k+ 1){-1/2\choose k}^2\equiv p\Phi_p(- 1)\pmod{p^3},\tag{\(*\)} \] where \(\Phi_p(x)\) is the Legendre symbol modulo \(p\).
Formula \((*)\) is the \(p\)-adic analog of Ramanujan’s formula: \[ \sum^\infty_{k=0} (4k+1){-1/2\choose k}^3= 2/\pi. \]

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
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[1] 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. · Zbl 1037.33016
[2] Scott Ahlgren and Ken Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187 – 212. · Zbl 0940.33002
[3] W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. · Zbl 0011.02303
[4] F. Beukers, Some congruences for the Apéry numbers, J. Number Theory 21 (1985), no. 2, 141 – 155. · Zbl 0571.10008
[5] F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201 – 210. · Zbl 0614.10011
[6] P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas, Calabi-Yau manifolds over finite fields I, http://xxx.lanl.gov/abs/hep-th/0012233. · Zbl 1100.14032
[7] John Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77 – 101. · Zbl 0629.12017
[8] Tsuneo Ishikawa, On Beukers’ conjecture, Kobe J. Math. 6 (1989), no. 1, 49 – 51. · Zbl 0687.10003
[9] Timothy Kilbourn, An extension of the Apéry number supercongruence, Acta Arith. 123 (2006), no. 4, 335 – 348. · Zbl 1170.11008
[10] D. McCarthy and R. Osburn, A p-adic analogue of a formula of Ramanujan, preprint. · Zbl 1175.33004
[11] Eric Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99 (2003), no. 1, 139 – 147. · Zbl 1074.11045
[12] Eric Mortenson, Supercongruences between truncated \(_{2}\)\?\(_{1}\) hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc. 355 (2003), no. 3, 987 – 1007. · Zbl 1074.11044
[13] Eric Mortenson, Supercongruences for truncated _{\?+1}\?_{\?} hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc. 133 (2005), no. 2, 321 – 330. · Zbl 1152.11327
[14] F. Morley, Note on the congruence \( 2^{4n}\equiv(-1)^n (2n)!/(n!)^2\), where \( 2n+1\) is prime, Annals of Math., 9 (1895), pages 168-170. · JFM 26.0208.02
[15] Ken Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1205 – 1223. · Zbl 0910.11054
[16] Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and \?-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. · Zbl 1119.11026
[17] Fernando Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 223 – 231. · Zbl 1062.11038
[18] E.T. Whittaker and G.N. Watson, Modern Analysis, fourth ed., Cambridge University Press, 1927. · JFM 53.0180.04
[19] L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, \?-adic functional analysis (Nijmegen, 1996) Lecture Notes in Pure and Appl. Math., vol. 192, Dekker, New York, 1997, pp. 223 – 236. · Zbl 0895.11051
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