| | |
| | |
Stat |
Members: 3667 Articles: 2'599'751 Articles rated: 2609
07 February 2025 |
|
| | | |
|
Article overview
| |
|
Linear transformations and strong $q$-log-concavity for certain combinatorial triangle | Bao-Xuan Zhu
; | Date: |
1 May 2016 | Abstract: | It is well-known that the binomial transformation preserves the log-concavity
property and log-convexity property. Let $inom{a+n}{b+k}$ be the binomial
coefficients and $inom{n,k}{j}$ be defined by
$(b_0+b_1x+cdots+b_kx^{k})^n:=sum_{j=0}^{kn}inom{n,k}{j}x^j,$ where the
sequence $(b_i)_{0leq ileq k}$ is log-concave. In this paper, we prove that
the linear transformation $$y_n(q)=sum_{k=0}^ninom{a+n}{b+k}x_k(q)$$
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 $$y_n=sum_{i=0}^{kn}inom{n,k}{j}x_i$$ 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 $[A_{n,k}(q)]_{n, kgeq0}$ be an infinite lower triangular array
of polynomials in $q$ with nonnegative coefficients satisfying the recurrence
egin{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 $ngeq 1$ and $kgeq 0$, 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 $[A_{n,k}(q)]_{n, kgeq0}$.
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. | Source: | arXiv, 1605.0257 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
|
| |
|
|
|