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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The limiting distribution of the coefficients of the $ q$-Catalan numbers


Authors: William Y. C. Chen, Carol J. Wang and Larry X. W. Wang
Journal: Proc. Amer. Math. Soc. 136 (2008), 3759-3767
MSC (2000): Primary 05A16, 60C05
Published electronically: July 3, 2008
MathSciNet review: 2425713
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Abstract: We show that the limiting distributions of the coefficients of the $ q$-Catalan numbers and the generalized $ q$-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small $ n$, we conjecture that for sufficiently large $ n$, the coefficients are unimodal and even log-concave except for a few terms of the head and tail.


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Additional Information

William Y. C. Chen
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: chen@nankai.edu.cn

Carol J. Wang
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: wangjian@cfc.nankai.edu.cn

Larry X. W. Wang
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: wxw@cfc.nankai.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09464-1
PII: S 0002-9939(08)09464-1
Keywords: Bernoulli number, $q$-Catalan number, unimodality, log-concavity, moment generating function
Received by editor(s): August 20, 2007
Published electronically: July 3, 2008
Additional Notes: The authors are grateful to the referee for valuable suggestions. Thanks are also due to Barbara Margolius and Helmut Prodinger for helpful comments. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
Communicated by: Jim Haglund
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.