Tauraso, Roberto \(q\)-analogs of some congruences involving Catalan numbers. (English) Zbl 1270.11016 Adv. Appl. Math. 48, No. 5, 603-614 (2012). 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 \([n+1]_q=(1-q^{n+1})/(1-q)\). Reviewer: Romeo Meštrović (Kotor) Cited in 2 ReviewsCited in 36 Documents MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities 11A07 Congruences; primitive roots; residue systems 05A10 Factorials, binomial coefficients, combinatorial functions 05A19 Combinatorial identities, bijective combinatorics Keywords:\(q\)-analogs; \(q\)-binomial coefficients; \(q\)-Catalan numbers; \(q\)-Fibonacci numbers; congruences; cyclotomic polynomials \(q\)-analogs; \(q\)-binomial coefficients; \(q\)-Catalan numbers; \(q\)-Fibonacci numbers; congruences; cyclotomic polynomials Citations:Zbl 0721.05001 PDFBibTeX XMLCite \textit{R. Tauraso}, Adv. Appl. Math. 48, No. 5, 603--614 (2012; Zbl 1270.11016) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and a Catalan number. Number of ways to express n as the sum of an odd prime, a Lucas number and a Catalan number. 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.