## 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, $${}_2F_1$$

supercongruences

Zbl 1062.11038
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### References:

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