Hankel-total positivity of some sequences

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 ≥ 0 (a_n)_{n\geq 0} is a Stieltjes moment sequence, then so is ( a n + 2 a n − a n + 1 2 ) n ≥ 0 (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 ( Γ ( n 0 + a i + 1 ) Γ ( k 0 + b i + 1 ) Γ ( ( n 0 − k 0 ) + ( a − b ) i + 1 ) ∏ j = 0 m 1 d j i + e j ) i ≥ 0 \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 ( ∑ k ≥ 0 α k λ k n ) n ≥ 0 \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 ( p n + r − 1 n ) \binom {pn+r-1}{n} and Fuss-Catalan numbers r p n + r ( p n + r n ) \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.