Zhai, Hongcun; He, Bing; Li, Long Some congruences on \(q\)-Franel numbers and \(q\)-Catalan numbers. (English) Zbl 1425.11033 Indian J. Pure Appl. Math. 50, No. 1, 181-189 (2019). Summary: We present certain \(q\)-analogues of Franel numbers and Catalan numbers, and establish several congruences on these \(q\)-numbers MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities Keywords:\(q\)-Franel numbers; \(q\)-Catalan numbers; \(q\)-congruence PDFBibTeX XMLCite \textit{H. Zhai} et al., Indian J. Pure Appl. Math. 50, No. 1, 181--189 (2019; Zbl 1425.11033) Full Text: DOI References: [1] G. E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, 1998. · Zbl 0996.11002 [2] N. J. Calkin, Factors of sums of powers of binomial coefficients, Acta Arith., 86 (1998), 17-26. · Zbl 0917.11011 · doi:10.4064/aa-86-1-17-26 [3] J. Franel, On a question of Laisant, L’Intermédiaire des Mathématiciens, 1 (1894), 45-47. [4] Victor J. W. Guo, Proof of two conjectures of Sun on congruences for Franel numbers, Integral Transforms Spec. Funct., 24 (2013), 532-539. · Zbl 1360.11006 · doi:10.1080/10652469.2012.715160 [5] Victor J. W. Guo and J. Zeng, New congruences for sums involving Apéry numbers or central Delannoy numbers, Int. J. Number Theory, 8 (2012), 2003-2016. · Zbl 1268.11028 · doi:10.1142/S1793042112501138 [6] Victor J. W. Guo and J. Zeng, Some congruences involving central q-binomial coefficients, Adv. in Appl. Math., 45 (2010), 303-316. · Zbl 1231.11020 · doi:10.1016/j.aam.2009.12.002 [7] F. Jarvis and H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J., 22 (2010), 171-186. · Zbl 1218.11025 · doi:10.1007/s11139-009-9218-5 [8] G. Olive, Generalized powers, Amer. Math. Monthly, 72 (1965), 619-627. · Zbl 0215.07003 · doi:10.1080/00029890.1965.11970580 [9] H. Pan, A q-analogue of Lehmer’s congruence, Acta Arith., 128 (2007), 303-318. · Zbl 1138.11005 · doi:10.4064/aa128-4-1 [10] Z.-W. Sun, Congruences for Franel numbers, Adv. in Appl. Math., 51 (2013), 524-535. · Zbl 1281.05007 · doi:10.1016/j.aam.2013.06.004 [11] Z.-W. Sun, Open conjectures on congruences, arXiv.org/abs/0911.5665. · Zbl 1449.11001 [12] Y. Zhang, Some congruences for Catalan numbers and binomial sums, Advances in Mathematics (China), 43 (2014), 857-862. · Zbl 1324.11021 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.