Yet another criterion for the total positivity of Riordan arrays

TL;DR

This paper introduces a new criterion for establishing the total positivity of Riordan arrays, showing that the positivity of a specific associated matrix implies the total positivity of the array.

Contribution

It provides a novel sufficient condition for the total positivity of Riordan arrays based on an associated matrix's properties.

Findings

The total positivity of the associated matrix guarantees the total positivity of the Riordan array.

The criterion simplifies verifying total positivity for classes of Riordan arrays.

The approach connects matrix positivity properties with combinatorial array structures.

Abstract

Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\ d_2 & h_2 & h_1 & h_0 &\\ \vdots&\vdots&&&\ddots \end{array}\right] \end{equation*} is totally positive, then so is the Riordan array $R$.