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Some \(q\)-analogues of supercongruences of Rodriguez-Villegas. (English) Zbl 1315.11015

Summary: We study different \(q\)-analogues and generalizations of the ex-conjectures of F. Rodriguez-Villegas [Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Fields Inst. Commun. 38, 223–231 (2003; Zbl 1062.11038)]. For example, for any odd prime \(p\), we show that the known congruence \(\sum_{k = 0}^{p - 1} \frac{\binom{2k} {k}^2}{16^k} \equiv(\frac{- 1}{p})\pmod {p^2}\), where \((\frac{\cdot}{p})\) is the Legendre symbol, has the following two nice \(q\)-analogues: \[ \sum_{k = 0}^{p - 1} \frac{(q; q^2)_k^2}{(q^2; q^2)_k^2} q^{(1 + \varepsilon) k} \equiv\left(\frac{- 1}{p}\right) q^{\frac{(p^2 - 1) \varepsilon}{4}}\pmod{(1 + q + \cdots + q^{p - 1})^2}, \] where \((a;q)_n = (1 - a)(1 - a q) \cdots(1 - a q^{n - 1})\) and \(\varepsilon = \pm 1\). Several related conjectures are also proposed.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
05A30 \(q\)-calculus and related topics

Citations:

Zbl 1062.11038
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References:

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