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Supercongruences between truncated \(_{2}F_{1}\) hypergeometric functions and their Gaussian analogs. (English) Zbl 1074.11044

F. Rodriguez-Villegas [Hypergeometric families of Calabi-Yau manifolds, Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 38, 223–231 (2003; Zbl 1062.11038)] conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension \(d\leq 3\). For manifolds of dimension \(d=1\), he observed four potential supercongruences. Later that author proved one of the four. Motivated by Rodriguez-Villegas’s work, in the paper under review the present author proves a general result on supercongruences between values of truncated \(_{2}F_{1}\) hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, he proves the three remaining supercongruences.

MSC:

11L10 Jacobsthal and Brewer sums; other complete character sums
11G20 Curves over finite and local fields
11A07 Congruences; primitive roots; residue systems
33C05 Classical hypergeometric functions, \({}_2F_1\)

Citations:

Zbl 1062.11038
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References:

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