Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity

Abstract

Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation
\[
T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1).
\]
In this paper, we give a new sufficient condition for linear transformations
\[
Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q)
\]
that preserves the strong $q$-log-convexity of polynomials sequences. As applications, we show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.