## 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

Zbl 1062.11038
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### References:

 [1] Ahlgren, S.; Ono, K., A Gaussian hypergeometric series evaluation and apéry number congruences, J. Reine Angew. Math., 518, 187-212, (2000) · Zbl 0940.33002 [2] Andrews, G. E., On the q-analog of Kummer’s theorem and applications, Duke Math. J., 40, 525-528, (1973) · Zbl 0266.33003 [3] Andrews, G. E., Applications of basic hypergeometric functions, SIAM Rev., 16, 441-484, (1974) · Zbl 0299.33004 [4] Andrews, G. E., The theory of partitions, (1998), Cambridge University Press Cambridge · Zbl 0996.11002 [5] Andrews, G. E., q-analogs of the binomial coefficient congruences of babbage, Wolstenholme and glaisher, Discrete Math., 204, 15-25, (1999) · Zbl 0937.05014 [6] Beukers, F., Another congruence for the apéry numbers, J. Number Theory, 25, 201-210, (1987) · Zbl 0614.10011 [7] Chan, K. K.; Long, L.; Zudilin, V. V., A supercongruence motivated by the Legendre family of elliptic curves, Mat. Zametki, Math. Notes, 88, 599-602, (2010), translation in: · Zbl 1252.11017 [8] Gasper, G.; Rahman, M., Basic hypergeometric series, Encyclopedia Math. Appl., vol. 96, (2004), Cambridge University Press Cambridge · Zbl 1129.33005 [9] Guo, V. J.W.; Zeng, J., Some congruences involving central q-binomial coefficients, Adv. in Appl. Math., 45, 303-316, (2010) · Zbl 1231.11020 [10] McCarthy, D.; Osburn, R., A p-adic analogue of a formula of Ramanujan, Arch. Math., 91, 492-504, (2008) · Zbl 1175.33004 [11] Mortenson, E., A supercongruence conjecture of rodriguez-villegas for a certain truncated hypergeometric function, J. Number Theory, 99, 139-147, (2003) · Zbl 1074.11045 [12] Mortenson, E., Supercongruences between truncated $${}_2F_1$$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc., 355, 987-1007, (2003) · Zbl 1074.11044 [13] Pan, H., A q-analogue of Lehmer’s congruence, Acta Arith., 128, 303-318, (2007) · Zbl 1138.11005 [14] Rodriguez-Villegas, F., Hypergeometric families of Calabi-Yau manifolds, (Calabi-Yau Varieties and Mirror Symmetry, Toronto, ON, 2001, Fields Inst. Commun., vol. 38, (2003), Amer. Math. Soc. Providence, RI), 223-231 · Zbl 1062.11038 [15] Shi, L.-L.; Pan, H., A q-analogue of Wolstenholme’s harmonic series congruence, Amer. Math. Monthly, 114, 529-531, (2007) · Zbl 1193.11018 [16] Straub, A., A q-analog of Ljunggren’s binomial congruence, (23rd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2011, Discrete Math. Theor. Comput. Sci. Proc., AO, (2011), Assoc. Discrete Math. Theor. Comput. Sci. Nancy), 897-902 · Zbl 1355.05053 [17] Sun, Z.-H., Congruences concerning Legendre polynomials, Proc. Amer. Math. Soc., 139, 1915-1929, (2011) · Zbl 1225.11006 [18] Sun, Z.-H., Generalized Legendre polynomials and related supercongruences, J. Number Theory, 143, 293-319, (2014) · Zbl 1353.11005 [19] Sun, Z.-W., Super congruences and Euler numbers, Sci. China Math., 54, 2509-2535, (2011) · Zbl 1256.11011 [20] Sun, Z.-W., On sums involving products of three binomial coefficients, Acta Arith., 156, 123-141, (2012) · Zbl 1269.11019 [21] Sun, Z.-W., Supercongruences involving products of two binomial coefficients, Finite Fields Appl., 22, 24-44, (2013) · Zbl 1331.11012 [22] Tauraso, R., An elementary proof of a rodriguez-villegas supercongruence, (2009), preprint [23] Tauraso, R., Supercongruences for a truncated hypergeometric series, Integers, 12, (2012), #A45 · Zbl 1301.11020 [24] Tauraso, R., Some q-analogs of congruences for central binomial sums, Colloq. Math., 133, 133-143, (2013) · Zbl 1339.11003 [25] Van Assche, W., Little q-Legendre polynomials and irrationality of certain Lambert series, Ramanujan J., 5, 295-310, (2001) · Zbl 1035.11032 [26] van Hamme, L., Some conjectures concerning partial sums of generalized hypergeometric series, (p-Adic Functional Analysis, Nijmegen, 1996, Lect. Notes Pure Appl. Math., vol. 192, (1997), Dekker), 223-236 · Zbl 0895.11051
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