Total positivity of recursive matrices

TL;DR

This paper establishes conditions under which certain recursively defined lower triangular matrices, including well-known combinatorial triangles, are totally positive, providing a unified framework for their analysis.

Contribution

It introduces sufficient conditions for total positivity of recursive matrices and applies these to classical combinatorial triangles.

Findings

Pascal, Stirling, Bell, and Catalan triangles are totally positive under the new conditions.

Unified approach simplifies proving total positivity for various combinatorial matrices.

Provides theoretical foundation for analyzing total positivity in recursive matrices.

Abstract

Let $A=[a_{n,k}]_{n,k\ge 0}$ be an infinite lower triangular matrix defined by the recurrence $$a_{0,0}=1,\quad a_{n+1,k}=r_{k}a_{n,k-1}+s_{k}a_{n,k}+t_{k+1}a_{n,k+1},$$ where $a_{n,k}=0$ unless $n\ge k\ge 0$ and $r_k,s_k,t_k$ are all nonnegative. Many well-known combinatorial triangles are such matrices, including the Pascal triangle, the Stirling triangle (of the second kind), the Bell triangle, the Catalan triangles of Aigner and Shapiro. We present some sufficient conditions such that the recursive matrix $A$ is totally positive. As applications we give the total positivity of the above mentioned combinatorial triangles in a unified approach.