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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hankel-total positivity of some sequences
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by Bao-Xuan Zhu PDF
Proc. Amer. Math. Soc. 147 (2019), 4673-4686 Request permission

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
  • MR Author ID: 902213
  • Email: bxzhu@jsnu.edu.cn
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4673-4686
  • MSC (2010): Primary 11B83, 15B05, 33B15, 05A20
  • DOI: https://doi.org/10.1090/proc/14599
  • MathSciNet review: 4011504