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.