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

Author(s): William Y. C. Chen; Carol J. Wang; Larry X. W. Wang
Journal: Proc. Amer. Math. Soc. 136 (2008), 3759-3767.
MSC (2000): Primary 05A16, 60C05
Posted: July 3, 2008
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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: 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
Posted: 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
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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