Stieltjes moment properties and continued fractions from combinatorial triangles

TL;DR

This paper investigates the Stieltjes moment properties and log-convexity of combinatorial triangle sequences defined by a recurrence, using continued fractions and linear transformations, with applications to various combinatorial numbers.

Contribution

It develops criteria for Stieltjes moment properties and log-convexity of row-generating functions of combinatorial triangles, extending understanding through continued fractions and transformations.

Findings

Criteria for x-Stieltjes moment property established

Linear transformations preserving moment properties demonstrated

Applications to factorial, Whitney, Stirling, and other combinatorial numbers

Abstract

Many 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 suitable $a_0,a_1,a_2,b_0,b_1,b_2,d$ and $\lambda$. For $n\geq0$, denote by $T_n(q)$ the generating function of the $n$-th row. In this paper, we develop various criteria for $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ based on the Jacobi continued fraction expression of $\sum_{n\geq0}T_n(q)t^n$, where $\textbf{x}$ is a set of indeterminates consisting of $q$ and those parameters occurring in the recurrence relation. With the help of a criterion of Wang and Zhu [Adv. in Appl. Math. (2016)], we show that the corresponding linear transformation of $T_{n,k}$ preserves Stieltjes moment properties of sequences. Finally, we present some related examples including factorial numbers, Whitney numbers, Stirling permutations, minimax trees and peak statistics.