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Congruences from $q$-Catalan Identities | ||
| Transactions on Combinatorics | ||
| مقاله 58، دوره 5، شماره 4، اسفند 2016، صفحه 57-67 اصل مقاله (225.98 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2016.20358 | ||
| نویسنده | ||
| Qing Zou* | ||
| Department of Mathematics, The University of Iowa | ||
| چکیده | ||
| In this paper, by studying three $q$-Catalan identities given by Andrews, we arrive at a certain number of congruences. These congruences are all modulo $\Phi_n(q)$, the $n$-th cyclotomic polynomial or the related functions and modulo $q$-integers. | ||
| کلیدواژهها | ||
| Congruences؛ $q$-Catalan identities؛ Catalan numbers؛ $q$-integer؛ Cyclotomic polynomial | ||
| مراجع | ||
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[1] C. A. Pickover, The Math Book: From Pythagoras to the 57th Dimension, 250, Milestones in the History of Mathematics, Sterling, New York, 2009. [2] T. Koshy, Catalan Numbers with Applications, Oxford Univ. Press, New York, 2009. [3] R. P. Stanley, Enumerative Combinatorics, 2, Cambridge Univ. Press, 1999. [4] R. P. Stanley, Catalan Numbers, Cambridge Univ. Press, 2015. [5] H. W. Gould, Bell and Catalan Numbers: Research Bibliography of Two Special Number Sequences, rev. ed., Com- binatorial Research Institute, Morgantown, WV, 1978. [6] J. Furlinger and J. Hofbauer, q-Catalan numbers, J. Combin. Theory Ser. A, 40 (1985) 248{264. [7] G. Gasper and M. Rahman, Basic Hypergeometric Series, second ed., Encyclopedia Math. Appl., 96, Cambridge Univ. Press, Cambridge, 2004. [8] G. E. Andrews, On the difference of successive Gaussian polynomials, J. Statist. Plann. Inference, 34 (1993) 19{22. [9] G. E. Andrews, Catalan numbers, q-Catalan numbers and hypergeometric series, J. Combin. Theory Ser. A, 44 (1987) 267{273. [10] G. E. Andrews, q-Catalan Identities, The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, (2010) 183{190. [11] R. Tauraso, q-Analogs of some congruences involving Catalan numbers, Adv. in Appl. Math., 48 (2012) 603{614. [12] W. E. Clark, q-Analogue of a binomial coefficient congruence, Internat. J. Math. Math. Sci., 18 (1995) 197{200. [13] E. Allen, Department of Mathematical Sciences, Carnegie Mellon University, Combinatorial Interpretations of Gen- eralizations of Catalan Numbers And Ballot Numbers, 2014, PhD Thesis. Available at: http://repository.cmu. edu/dissertations/366 [14] G.E. Andrews, The Theory of Partitions, Encyclopedia Math. Appl., 2, Addison-Wesley, Reading, MA, 1976. [15] E. Catalan, Question 1135, Nouvelles Annales de Mathematiques, Journal des Candidats aux Ecole Polytechnic et Normale, Series 2, 13 (1874) pp. 207. [16] G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev., 16 (1974) 441{484. | ||
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