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Proof of a \(q\)-congruence conjectured by Tauraso. (English) Zbl 1467.11027

This paper marks another milestone on the flourishing “binomial road” between China and Italy.
In fact, after having recalled the \(q\)-shifted factorials \((a;q)_n\), the \(q\)-binomial coefficients \(\left[\begin{matrix} n\\ k\end{matrix}\right]\), and the \(n\)th cyclotomic polynomials \(\Phi_n(q)\), the author proves the following theorem \[ \sum_{k=0}^{n-1} \frac{q^k}{(-q;q)_k} \left[\begin{matrix} 2k\\ k\end{matrix}\right] \equiv (-1)^{\frac{n-1}{2}} q^{\frac{n^2-1}{4}} \pmod{\Phi_n(q)^2}, \] which was conjectured by R. Tauraso [Colloq. Math. 133, No. 1, 133–143 (2013; Zbl 1339.11003)] while exploring the congruence \[ \sum_{k=0}^{n-1} \frac{q^k}{(-q;q)_k} \left[\begin{matrix} 2k\\ k\end{matrix}\right] \equiv (-1)^{\frac{n-1}{2}} q^{\frac{n^2-1}{4}} \pmod{\Phi_n(q)}, \] established by V. J. W. Guo and J. Zeng [Adv. Appl. Math. 45, No. 3, 303–316 (2010; Zbl 1231.11020)] as \(q\)-analogue of \[ \sum_{k=0}^{p^r-1} \frac{1}{2^k} \binom{2k}{k} \equiv (-1)^{\frac{p^r-1}{2}} \pmod{p}\] due to Z.-W. Sun and R. Tauraso [Adv. Appl. Math. 45, No. 1, 125–148 (2010; Zbl 1231.11021)].
The proof employs some properties of the \(q\)-WZ pair, the Carlitz’s transformation formula, a \(q\)-analogue of Morley’s congruence supplied by J. Liu et al. [Adv. Difference Equ. 2015, Paper No. 254, 7 p. (2015; Zbl 1422.11049)], and an application of the \(q\)-Chu-Vandermonde summation formula given by the author [J. Math. Anal. Appl. 458, No. 1, 590–600 (2018; Zbl 1373.05025)].
As auxiliary results, the author finds the following identity \[\sum_{k=0}^{n-1} \frac{q^k}{(-q;q)_k } \left[\begin{matrix} 2k\\ k\end{matrix}\right] = \sum_{k=1}^{n} (-1)^{k-1} \frac{(q;q^2)_n q^{k^2-k}}{(q;q)_{n-k} (q;q^2)_k},\] and he conjectures three new congruences, two of which extend previous ones provided by the same Guo and Zeng as \(q\)-analogues of the congruence \[ \sum_{k=0}^{p^a-1} \binom{2k}{k} \equiv \left(\frac{p^a}{3}\right) \pmod{p^2}\] still from Z. Sun and R. Tauraso [Int. J. Number Theory 7, No. 3, 645–662 (2011; Zbl 1247.11027)].

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
05A30 \(q\)-calculus and related topics
05A10 Factorials, binomial coefficients, combinatorial functions
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References:

[1] Carlitz, L., A \(q\)-identity, Fibonacci Quart., 12, 369-372, (1974) · Zbl 0296.33001
[2] Gasper, G.; Rahman, M., Basic Hypergeometric Series, 96, (2004), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1129.33005
[3] Guo, V. J. W., A \(q\)-analogue of a Ramanujan-type supercongruence involving central binomial coefficients, J. Math. Anal. Appl., 458, 590-600, (2018) · Zbl 1373.05025
[4] Guo, V. J. W.; Zeng, J., Some congruences involving central \(q\)-binomial coefficients, Adv. Appl. Math., 45, 303-316, (2010) · Zbl 1231.11020
[5] V. J. W. Guo and W. Zudilin, A \(q\)-microscope for supercongruences, preprint (2018); arXiv:1803.01830.
[6] Liu, J.; Pan, H.; Zhang, Y., A generalization of Morley’s congruence, Adv. Differential Equation, 2015, 254, (2015)
[7] Sun, Z.-W., Binomial coefficients, Catalan numbers and Lucas quotients, Sci. China Math., 53, 2473-2488, (2010) · Zbl 1221.11054
[8] Sun, Z.-W.; Tauraso, R., New congruences for central binomial coefficients, Adv. Appl. Math., 45, 125-148, (2010) · Zbl 1231.11021
[9] Sun, Z.-W.; Tauraso, R., On some new congruences for binomial coefficients, Int. J. Number Theory, 7, 645-662, (2011) · Zbl 1247.11027
[10] Tauraso, R., Some \(q\)-analogs of congruences for central binomial sums, Colloq. Math., 133, 133-143, (2013) · Zbl 1339.11003
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