Total positivity of recursive matrices

Abstract

Let $A=[a_{n,k}{]}_{n,k\ge 0}$ be an infinite lower triangular matrix defined by the recurrence $$a_{0,0}=1,a_{n+1,k}=r_k{a}_{n,k-1}+s_k{a}_{n,k}+t_{k+1}{a}_{n,k+1},$$ 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.

References in corpus (1)

• Total positivity of Riordan arrays

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• Positivity properties of some special matrices
• Catalan-like numbers and Hausdorff moment sequences
• Yet another criterion for the total positivity of Riordan arrays
• Optimal interval length for the collocation of the Newton basis
• On a Stirling-Whitney-Riordan triangle
• Combinatorial identities related to $2\times 2$ submatrices of recursive matrices
• Stieltjes moment properties and continued fractions from combinatorial triangles
• Total positivity from the exponential Riordan arrays