On a Stirling-Whitney-Riordan triangle

TL;DR

This paper introduces a new Stirling-Whitney-Riordan triangle, proves its total positivity, real-rootedness, and stability properties, and provides explicit formulas and continued fraction expansions, advancing combinatorial and algebraic understanding of these triangles.

Contribution

It establishes the total positivity, real zeros, and stability of the Stirling-Whitney-Riordan triangle, along with explicit formulas and generating functions, extending known properties of related combinatorial triangles.

Findings

The triangle is x-totally positive.

Row-generating functions have only real zeros.

The triangle exhibits stability and log-convexity properties.

Abstract

Based on the Stirling triangle of the second kind, the Whitney triangle of the second kind and one triangle of Riordan, we study a Stirling-Whitney-Riordan triangle $[T_{n,k}]_{n,k}$ satisfying the recurrence relation: \begin{eqnarray*} T_{n,k}&=&(b_1k+b_2)T_{n-1,k-1}+[(2\lambda b_1+a_1)k+a_2+\lambda( b_1+b_2)] T_{n-1,k}+\\ &&\lambda(a_1+\lambda b_1)(k+1)T_{n-1,k+1}, \end{eqnarray*} where initial conditions $T_{n,k}=0$ unless $0\le k\le n$ and $T_{0,0}=1$. We prove that the Stirling-Whitney-Riordan triangle $[T_{n,k}]_{n,k}$ is $\textbf{x}$-totally positive with $\textbf{x}=(a_1,a_2,b_1,b_2,\lambda)$. We show that the row-generating function $T_n(q)$ has only real zeros and the Tur\'{a}n-type polynomial $T_{n+1}(q)T_{n-1}(q)-T^2_n(q)$ is stable. We also present explicit formulae for $T_{n,k}$ and the exponential generating function of $T_n(q)$ and give a Jacobi continued fraction expansion for the ordinary generating function of $T_n(q)$. Furthermore, we get the $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ and show that the triangular convolution $z_n=\sum_{i=0}^nT_{n,i}x_iy_{n-i}$ preserves Stieltjes moment property of sequences. Finally, for the first column $(T_{n,0})_{n\geq0}$, we derive some properties similar to those of $(T_n(q))_{n\geq0}.$