Total Positivity of Almost-Riordan Arrays

TL;DR

This paper investigates the total positivity of almost-Riordan arrays, establishing necessary and sufficient conditions, and explores the role of production matrices, including counterexamples and specific cases with tridiagonal matrices.

Contribution

It provides a comprehensive characterization of total positivity for almost-Riordan arrays, linking it to production matrices and introducing new conditions and examples.

Findings

Total positivity characterized by sequence conditions.

Production matrix positivity implies array positivity.

Counterexample shows conditions are not always necessary.

Abstract

In this paper we study the total positivity of almost-Riordan arrays $(d(t)|\, g(t), f(t))$ and establish its necessary conditions and sufficient conditions, particularly, for some well used formal power series $d(t)$. We present a semidirect product of an almost-array and use it to transfer a total positivity problem for an almost-Riordan array to the total positivity problem for a quasi-Riordan array. We find the sequence characterization of total positivity of the almost-Riordan arrays. The production matrix $J$ of an almost-Riordan array $(d|\, g,f)$ is presented so that $J$ is totally positive implies the total positivity of both the almost-Riordan array $(d|\, g,f)$ and the Riordan array $(g,f)$. We also present a counterexample to illustrate that this sufficient condition is not necessary. If the production matrix $J$ is tridiagonal, then the expressions of its principal minors are given. By using expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. A numerous examples are given to demonstrate our results.