Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Linear transformations and strong $q$-log-concavity for certain
  combinatorial triangle   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.

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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph