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Hankel-total positivity of some sequences


Author: Bao-Xuan Zhu
Journal: Proc. Amer. Math. Soc. 147 (2019), 4673-4686
MSC (2010): Primary 11B83, 15B05, 33B15, 05A20
DOI: https://doi.org/10.1090/proc/14599
Published electronically: May 29, 2019
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Abstract: The aim of this paper is to develop analytic techniques to deal with Hankel-total positivity of sequences.

We show two nonlinear operators preserving Stieltjes moment property of sequences. They actually both extend a result of Wang and Zhu that if $ (a_n)_{n\geq 0}$ is a Stieltjes moment sequence, then so is $ (a_{n+2}a_{n}-a^2_{n+1})_{n\geq 0}$. Using complete monotonicity of functions, we also prove Stieltjes moment properties of the sequences $ \left ( \frac {\Gamma (n_{0}+ai+1)}{{\Gamma (k_{0}+bi+1)} {\Gamma ((n_0-k_0)+(a-b)i+1)}}\prod _{j=0}^m\frac {1}{d_ji+e_j}\right )_{i\geq 0}$ and $ \left (\sum _{k\ge 0}\frac {\alpha _k}{\lambda _{k}^{n}}\right )_{n\geq 0}$. Particularly in a new unified manner our results imply the Stieltjes moment properties of binomial coefficients $ \binom {pn+r-1}{n}$ and Fuss-Catalan numbers $ \frac {r}{pn+r}\binom {pn+r}{n}$ proved by Mlotkowski, Penson, and Zyczkowski, and Liu and Pego, respectively, and also extend some results for log-convexity of sequences proved by Chen-Guo-Wang, Su-Wang, Yu, and Wang-Zhu, respectively.


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Additional Information

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

DOI: https://doi.org/10.1090/proc/14599
Keywords: Total positivity, Hankel matrices, Stieltjes moment property, complete monotonicity
Received by editor(s): September 17, 2018
Received by editor(s) in revised form: February 19, 2019, and February 25, 2019
Published electronically: May 29, 2019
Additional Notes: The author was partially supported by the National Natural Science Foundation of China (No. 11571150).
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society