Some positivities in certain triangular arrays

Zhu, Bao-Xuan

doi:10.1090/S0002-9939-2014-12008-9

Some positivities in certain triangular arrays
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Proc. Amer. Math. Soc. 142 (2014), 2943-2952 Request permission

Abstract:

Let $\{T_{n,k}\}_{n,k\ge 0}$ be an array of nonnegative numbers satisfying the recurrence relation \begin{equation*} T_{n,k}=(a_1k^2+a_2k+a_3)T_{n-1,k}+(b_1k^2+b_2k+b_3)T_{n-1,k-1} \end{equation*} with $T_{n,k}=0$ unless $0\le k\le n$. We obtain some results for the total positivity of the matrix $\left (T_{n,k}\right )_{n,k\ge 0}$, Pólya frequency properties of the row and column generating functions, and $q$-log-convexity of the row generating functions. This allows a unified treatment of the properties above for some triangular arrays of the second kind, including the Stirling triangle, Jacobi-Stirling triangle, Legendre-Stirling triangle, and central factorial numbers triangle.

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