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\(q\)-analogs of some congruences involving Catalan numbers. (English) Zbl 1270.11016

In this paper, the author presents some variations on the Greene-Krammer’s identity and related congruences which involve \(q\)-Catalan numbers. In particular, by Theorem 4.2, for all integers \(n\) and \(d\) such that \(n\geq |d|\) we have \[ \sum_{k=0}^{n-1}q^k{2k \brack k+d}_q=\sum_{k=0}^{n-|d|} q^{\frac{1}{3}(2(n-k)^2-(n-k)\left(\frac{n-|d|-k}{3}\right)-2d^2-1)} \left(\frac{n-|d|-k}{3}\right){2n \brack k}_q, \] where \({m \brack k}_q\) is the Gaussian \(q\)-binomial coefficient, and \(\left(\frac{a}{3}\right)\) is the Jacobi symbol which coincides with the unique integer \(\varepsilon\in\{-1,0,1\}\) satisfying \(a\equiv\varepsilon \pmod 3\).
As an application of the above identity, Corollary 4.3 asserts that for \(n\geq |d|\) the left hand side of the above identity is \(\equiv \left(\frac{n-|d|}{3}\right)q^{\frac{3}{2}r(r+1)+|d|(2r+1)}\pmod{\Phi_n(q)}\), where \(r=\lfloor 2(n-|d|)/3\rfloor\) and \(\Phi_n(q)\) is the \(n\)th cyclotomic polynomial.
Furthermore, the author established some identities and congruences modulo \(\Phi_n(q)\) which involve \(q\)-Catalan numbers \(C_n^q:=\frac{1}{[n+1]_q}{2n \brack n}_q={2n \brack n}_q-q{2n \brack n+1}_q\), where \([n+1]_q=(1-q^{n+1})/(1-q)\).

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics

Citations:

Zbl 0721.05001
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References:

[1] Andrews, G. E., The Theory of Partitions, Encyclopedia Math. Appl., vol. 2 (1976), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0371.10001
[2] Andrews, G. E., On the Greene-Krammer theorem and related identities, Ganita, 43, 69-73 (1992) · Zbl 0844.11020
[3] Carlitz, L., Fibonacci notes 4: \(q\)-Fibonacci polynomials, Fibonacci Quart., 13, 97-102 (1975) · Zbl 0298.10010
[4] Clark, W. E., \(q\)-Analogue of a binomial coefficient congruence, Int. J. Math. Math. Sci., 18, 197-200 (1995) · Zbl 0816.05009
[5] Chen, W. Y.C.; Hou, Q. H., Factors of the Gaussian coefficients, Discrete Math., 306, 1146-1449 (2006) · Zbl 1096.05008
[6] Chen, K. J.; Srivastava, H. M., A generalization of two \(q\)-identities of Andrews, J. Combin. Theory Ser. A, 95, 381-386 (2001) · Zbl 0990.11010
[7] Ekhad, S. B.; Zeilberger, D., The number of solutions of \(X^2 = 0\) in triangular matrices over \(GF(q)\), Electron. J. Combin., 3, R2 (1996), 1-2
[8] Fürlinger, J.; Hofbauer, J., \(q\)-Catalan numbers, J. Combin. Theory Ser. A, 40, 248-264 (1985) · Zbl 0581.05006
[9] Greene, J., On a conjecture of Krammer, J. Combin. Theory Ser. A, 56, 309-311 (1991) · Zbl 0721.05001
[10] Guo, V. J.W.; Zeng, J., Some congruences involving central \(q\)-binomial coefficients, Adv. in Appl. Math., 45, 303-316 (2010) · Zbl 1231.11020
[11] Nagell, T., Introduction to Number Theory (1951), Wiley: Wiley New York · Zbl 0042.26702
[12] Pan, H.; Sun, Z. W., A combinatorial identity with application to Catalan numbers, Discrete Math., 306, 1921-1940 (2006) · Zbl 1221.11052
[13] Sun, Z. W.; Tauraso, R., On some new congruences for binomial coefficients, Int. J. Number Theory, 7, 645-662 (2011) · Zbl 1247.11027
[14] Sun, Z. W.; Tauraso, R., New congruences for central binomial coefficients, Adv. in Appl. Math., 45, 125-148 (2010) · Zbl 1231.11021
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