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A \(q\)-microscope for supercongruences. (English) Zbl 1464.11028
Summary: By examining asymptotic behavior of certain infinite basic (\(q\)-) hypergeometric sums at roots of unity (that is, at a ‘\(q\)-microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a \(q\)-analogue of Ramanujan’s formula \[ \sum_{n = 0}^\infty \frac{\binom{4n}{2n}\binom{2n}{n}^2}{2^{8n} 3^{2n}}(8n + 1) = \frac{2 \sqrt{3}}{\pi}, \] of the two supercongruences \[ \begin{aligned} & S(p - 1) \equiv p\left(\frac{- 3}{p}\right)\pmod{p^3} \text{ and} \\ & S\left(\frac{p - 1}{2}\right) \equiv p\left(\frac{- 3}{p}\right)\pmod{p^3}, \end{aligned} \] valid for all primes \(p > 3\), where \(S(N)\) denotes the truncation of the infinite sum at the \(N\)-th place and \((\frac{- 3}{\cdot})\) stands for the quadratic character modulo 3.

MSC:
11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
11Y60 Evaluation of number-theoretic constants
33C20 Generalized hypergeometric series, \({}_pF_q\)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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