Linear transformations and strong $q$-log-concavity for certain combinatorial triangle

Abstract

It is well-known that the binomial transformation preserves the log-concavity property and log-convexity property. Let $\binom{a+n}{b+k}$ be the binomial coefficients and $\binom{n,k}{j}$ be defined by $(b_0+b_1x+\cdots+b_kx^{k})^n:=\sum_{j=0}^{kn}\binom{n,k}{j}x^j,$ where the sequence $(b_i)_{0\leq i\leq k}$ is log-concave. In this paper, we prove that the linear transformation $$y_n(q)=\sum_{k=0}^n\binom{a+n}{b+k}x_k(q)$$ preserves the strong $q$-log-concavity property for any fixed nonnegative integers $a$ and $b$, which strengthens and gives a simple proof of results of Ehrenborg and Steingrimsson, and Wang, respectively, on linear transformations preserving the log-concavity property. We also show that the linear transformation $$y_n=\sum_{i=0}^{kn}\binom{n,k}{j}x_i$$ not only preserves the log-concavity property, but also preserves the log-convexity property, which extends the results of Ahmia and Belbachir about the $s$-triangle transformation preserving the log-convexity property and log-concavity property. Let $[A_{n,k}(q)]_{n, k\geq0}$ be an infinite lower triangular array of polynomials in $q$ with nonnegative coefficients satisfying the recurrence \begin{eqnarray*}\label{re} A_{n,k}(q)=f_{n,k}(q)\,A_{n-1,k-1}(q)+g_{n,k}(q)\,A_{n-1,k}(q)+h_{n,k}(q)\,A_{n-1,k+1}(q), \end{eqnarray*} for $n\geq 1$ and $k\geq 0$, where $A_{0,0}(q)=1$, $A_{0,k}(q)=A_{0,-1}(q)=0$ for $k>0$. We present criterions for the strong $q$-log-concavity of the sequences in each row of $[A_{n,k}(q)]_{n, k\geq0}$. As applications, we get the strong $q$-log-concavity or the log-concavity of the sequences in each row of many well-known triangular arrays, such as the Bell polynomials triangle, the Eulerian polynomials triangle and the Narayana polynomials triangle in a unified approach.