A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function. (English) Zbl 1074.11045
Summary: F. Rodriguez-Villegas [Hypergeometric families of Calabi-Yau manifolds. Fields Inst. Commun. 38, 223–231 (2003; Zbl 1062.11038)] studied hypergeometric families of Calabi-Yau manifolds and found (numerically) many possible supercongruences. For example, he conjectured for every odd prime \(p\) that
\[ \sum_{n=0}^{p-1}\binom{2n}{n}^2 16^{-n}\equiv \left(\frac{-4}{p}\right)\pmod {p^2}. \]
Here, the author uses the theory of Gaussian hypergeometric series, the properties of the \(p\)-adic \(\Gamma\)-function, and a strange combinatorial identity to prove this conjecture.
In the paper reviewed above [Trans. Am. Math. Soc. 355, No. 3, 987–1007 (2003; Zbl 1074.11044)], the author has proved the other three congruences.
\[ \sum_{n=0}^{p-1}\binom{2n}{n}^2 16^{-n}\equiv \left(\frac{-4}{p}\right)\pmod {p^2}. \]
Here, the author uses the theory of Gaussian hypergeometric series, the properties of the \(p\)-adic \(\Gamma\)-function, and a strange combinatorial identity to prove this conjecture.
In the paper reviewed above [Trans. Am. Math. Soc. 355, No. 3, 987–1007 (2003; Zbl 1074.11044)], the author has proved the other three congruences.
MSC:
| 11A07 | Congruences; primitive roots; residue systems |
| 11G20 | Curves over finite and local fields |
| 11T24 | Other character sums and Gauss sums |
| 11L10 | Jacobsthal and Brewer sums; other complete character sums |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 33E50 | Special functions in characteristic \(p\) (gamma functions, etc.) |
References:
| [1] | Ahlgren, S., Gaussian hypergeometric series and combinatorial congruences, (Garvan, F. G.; Ismail, M. E.H., Developments in Mathematics, vol. 4 (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, The Netherlands), 1-12 · Zbl 1037.33016 |
| [2] | Ahlgren, S.; Ono, K., A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Ange. Math., 518, 187-212 (2000) · Zbl 0940.33002 |
| [3] | Beukers, F., Another Congruence for the Apéry Numbers, J. Number Theory, 25, 201-210 (1987) · Zbl 0614.10011 |
| [4] | Greene, J., Hypergeometric functions over finite fields, Trans. Amer. Math. Soc., 301, 77-101 (1987) · Zbl 0629.12017 |
| [5] | Gross, B.; Koblitz, N., Gauss sums and the \(p\)-adic \(Γ\)-function, Ann. Math., 109, 569-581 (1979) · Zbl 0406.12010 |
| [6] | Ishikawa, T., On Beurkers’ congruence, Kobe J. Math., 6, 49-52 (1989) |
| [7] | Kaar, M., Summation in finite terms, J. Assoc. Comput. Mach., 28, 2, 305-350 (1981) · Zbl 0494.68044 |
| [8] | Schneider, C., An implementation of Karr’s summation algorithm in mathematica, Sém. Lothar. Combin., S43b, 1-10 (1999) · Zbl 0941.68162 |
| [9] | F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, preprint.; F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, preprint. · Zbl 1062.11038 |
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