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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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