## 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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