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

 

Some positivities in certain triangular arrays


Author: Bao-Xuan Zhu
Journal: Proc. Amer. Math. Soc. 142 (2014), 2943-2952
MSC (2010): Primary 05A20
Published electronically: April 11, 2014
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Abstract: Let $ \{T_{n,k}\}_{n,k\ge 0}$ be an array of nonnegative numbers satisfying the recurrence relation

$\displaystyle T_{n,k}=(a_1k^2+a_2k+a_3)T_{n-1,k}+(b_1k^2+b_2k+b_3)T_{n-1,k-1}$    

with $ T_{n,k}=0$ unless $ 0\le k\le n$. We obtain some results for the total positivity of the matrix $ \left (T_{n,k}\right )_{n,k\ge 0}$, Pólya frequency properties of the row and column generating functions, and $ q$-log-convexity of the row generating functions. This allows a unified treatment of the properties above for some triangular arrays of the second kind, including the Stirling triangle, Jacobi-Stirling triangle, Legendre-Stirling triangle, and central factorial numbers triangle.

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Additional Information

Bao-Xuan Zhu
Affiliation: School of Mathematical Sciences, Jiangsu Normal University, Xuzhou 221116, People’s Republic of China
Email: bxzhu@jsnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12008-9
PII: S 0002-9939(2014)12008-9
Keywords: Recurrence relations, Jacobi-Stirling numbers, total positivity, log-concavity, P\'olya frequency, $q$-log-convexity
Received by editor(s): May 28, 2012
Received by editor(s) in revised form: August 27, 2012
Published electronically: April 11, 2014
Additional Notes: This work was partially supported by the National Natural Science Foundation of China (Nos. 11071030, 11201191), the Natural Science Foundation of Jiangsu Higher Education Institutions (No. 12KJB110005), Key Project of Chinese Ministry of Education (No. 212098), PAPD of Jiangsu Higher Education Institutions and Natural Science Foundation of Jiangsu Normal University (No. 11XLR30)
Communicated by: Jim Haglund
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.