Hankel-total positivity of some sequences

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.