Stieltjes moment properties and continued fractions from combinatorial triangles

Many famous combinatorial numbers can be placed in the following generalized triangular array $[T_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} T_{n,k}=\lambda(a_0n+a_1k+a_2)T_{n-1,k}+(b_0n+b_1k+b_2)T_{n-1,k-1}+\frac{d(da_1-b_1)}{\lambda}(n-k+1)T_{n-1,k-2} \end{equation*} with $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$. For $n\geq0$, denote by $T_n(q)$ its row-generating functions. In this paper, we consider the $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $(T_n(q))_{n\geq0}$ and the linear transformation of $T_{n,k}$ preserving Stieltjes moment properties of sequences. Using total positivity, we develop various criteria for $\textbf{x}$-Stieltjes moment property and $r$-$\textbf{x}$-log-convexity based on a four-term recursive array and Jacobi continued fraction expressions of generating functions. We apply our criteria to the $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ after we get the Jacobi continued fraction expression of $\sum_{n\geq0}T_n(q)t^n$. With the help of a criterion of Wang and Zhu [Adv. in Appl. Math. (2016)], we show that the corresponding linear transformation of $T_{n,k}$ preserve Stieltjes moment properties of sequences. Finally, we present some related famous examples including factorial numbers, Whitney numbers, Stirling permutations, minimax trees and peak statistics.