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\title{\bf Linear transformations preserving\\ the strong $q$-log-convexity of polynomials
}
\date{\dateline{Apr 5, 2015}{Jul 25, 2015}{Aug 28, 2015}\\
\small Mathematics Subject Classifications: 05A20; 15A15}

\author{Bao-Xuan Zhu\thanks{Supported partially by the National Natural Science Foundation of China (Grant No.
11201191).}\\
\small School of Mathematics and Statistics\\[-0.8ex]
\small Jiangsu Normal University\\[-0.8ex]
\small Xuzhou, PR China\\
\small\tt bxzhu@jsnu.edu.cn\\
\and
Hua Sun\\
\small College of Sciences\\[-0.8ex]
\small Dalian Ocean University\\[-0.8ex]
\small Dalian, PR China\\
\small\tt sunhua@dlou.edu.cn}

\begin{document}

\maketitle

\begin{abstract}
In this paper, we give a sufficient condition for the linear
transformation preserving the strong $q$-log-convexity. As
applications, we get some linear transformations (for instance,
Morgan-Voyce transformation, binomial transformation, Narayana
transformations of two kinds) preserving the strong
$q$-log-convexity. In addition, our results not only extend some
known results, but also imply the strong $q$-log-convexity of some
sequences of polynomials.

\bigskip\noindent \textbf{Keywords:} Log-concavity; Log-convexity;
$q$-Log-convexity; Strong $q$-log-convexity
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\hspace*{\parindent}
Let $a_0,a_1,a_2,\ldots$ be a sequence of nonnegative real numbers.
The sequence is called {\it log-concave} (resp. {\it log-convex}) if
for all $k\ge 1$, $a_{k-1}a_{k+1}\le a_k^2$ (resp.
$a_{k-1}a_{k+1}\ge a_k^2$). The log-concave sequences arise often in
combinatorics, algebra, geometry, analysis, probability and
statistics and have been extensively investigated. We refer the
reader to \cite{Sta89,Bre94, WY07} for log-concavity and
\cite{LW07,Zhu13} for log-convexity.

For a polynomial $f(q)$ with real coefficients, denote $f(q)\geq_q
0$ if it has only nonnegative coefficients. For a sequence of
polynomials with nonnegative coefficients $\{f_n(q)\}_{n\geq 0}$, it
is called {\it $q$-log-convex}, introduced by Liu and
Wang~\cite{LW07}, if
\begin{equation}\label{q-lcx}
\mathcal {T}(f_n(q))= f_{n+1}(q)f_{n-1}(q)- f_n(q)^2\geq_q
0\end{equation} for $n\geq 1$. If the opposite inequality in
(\ref{q-lcx}) holds, then it is called {\it $q$-log-concave}, first
suggested by Stanley. It is called {\it strongly $q$-log-convex} if
$$f_{n+1}(q)f_{m-1}(q)- f_n(q)f_m(q)\geq_q 0$$
for any $n\geq m\geq1$, see Chen {\it et al.} \cite{CWY11}. Clearly,
their strong $q$-log-convexity of polynomial sequences implies the
$q$-log-convexity. However, the converse does not hold, see Chen
{\it et al.} \cite{CWY11}. It is easy to see that if the sequence
$\{f_n(q)\}_{n\ge 0}$ is $q$-log-convex, then for each fixed
nonnegative number $q$, the sequence $\{f_n(q)\}_{n\ge 0}$ is
log-convex. The $q$-log-concavity and $q$-log-convexity of
polynomials have been extensively studied,
see~\cite{But90,BF07,CTWY10,CWY10,CWY11,DR13,DR132,Le90,LZ14,LW07,Sag92,Sag921,SWY11,Zhu13,Zhu14}
for instance.

It is a good way to obtain the log-concavity or log-convexity by
some operators. For instance, Davenport and P\'olya~\cite{DP49}
demonstrated that the log-convexity is preserved under the binomial
convolution. Wang and Yeh~\cite{WY07} also proved that the
log-concavity is preserved under the binomial convolution.
Br\"and\'en \cite{Bra06} studied some linear transformations
preserving the P\'olya frequency property of sequences. Liu and Wang
\cite{LW07} also studied the linear transformation preserving the
log-convexity. However, there is no result about the linear
transformation preserving the strong $q$-log-convexity. This is our
motivation of this paper.

It has been found that many polynomials have the strong
$q$-log-convexity. Let polynomials $\A_n(q)=\sum_{k=0}^na(n,k)q^k$
for $n\geq0$. Note that $\{q^k\}_{k\geq0}$ is a strongly
$q$-log-convex sequence. Thus it is natural to consider the strong
$q$-log-convexity of
\begin{equation*}\label{q-p}
\B_n(q)=\sum_{k=0}^na(n,k)f_k(q),
\end{equation*}
for $n\geq0$ if $\{f_n(q)\}_{n\geq0}$ is a strongly $q$-log-convex
sequence.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The objective of this paper is to study the strong $q$-log-convexity of
$\{\B_n(q)\}_{n\geq0}$. In Section 2, we first give a sufficient
condition implying the strong $q$-log-convexity of $\B_n(q)$, see
Theorem \ref{thm+s-q-general}. Then we apply it to some special
linear transformations. As consequences, on the one hand, we extend
some known results. On the other hand, we also get some new results on the strong
$q$-log-convexity of some sequences of polynomials.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Strong $q$-log-convexity and linear transformations}
\hspace*{\parindent}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given a triangular array $\{a(n,k)\}_{0\le k\le n}$ of nonnegative
real numbers and a strongly $q$-log-convex sequence
$\{f_n(q)\}_{n\geq0}$, define the polynomials
\begin{equation*}\label{q-p}
\A_n(q)=\sum_{k=0}^na(n,k)q^ k \quad\text{and}\quad
\B_n(q)=\sum_{k=0}^na(n,k)f_k(q),
\end{equation*}
for $n\geq0$. For convenience, let $a(n,k)=0$ unless $0\le k\le n$.
Suppose $m\geq n$. For $0\le t\le m+n$, define
\begin{align*}
    a_k(m,n,t)=&a(n-1,k)a(m+1,t-k)+a(m+1,k)a(n-1,t-k)\\ &-a(m,k)a(n,t-k)-a(n,k)a(m,t-k)
\end{align*}
if $0\le k<t/2$, and
\begin{equation*}
    a_k(m,n,t)=a(n-1,k)a(m+1,k)-a(n,k)a(m,k)
\end{equation*}
if $t$ is even and $k=t/2$. Our main result of this paper is as follows.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}\label{thm+s-q-general}
Suppose that the triangle $\{a(n,k)\}$ of nonnegative real numbers
satisfies the following two conditions:
\begin{itemize}
  \item [\rm(C1)] The sequence of polynomials $\{\A_n(q)\}_{n\ge 0}$ is strongly $q$-log-convex.
  \item [\rm(C2)] There exists an index $r=r(m,n,t)$ such that $a_k(m,n,t)\ge
  0$ for $k\le r$ and $a_k(m,n,t)<0$ for $k>r$.
\end{itemize}
If the sequence $\{f_k(q)\}_{k\ge 0}$ is strongly $q$-log-convex,
then the polynomials $\B_n(q)$ form a
strongly $q$-log-convex sequence.
\end{thm}

\begin{proof}
Let $m\geq n\geq0$. By computation, we have
$$\A_{n-1}(q)\A_{m+1}(q)-\A_n(q)\A_m(q)=\sum_{t=0}^{m+n}\left[\sum_{k=0}^{\lrf{t/2}}a_k(m,n,t)\right]q^t,$$
and
$$\B_{n-1}(q)\B_{m+1}(q)-\B_n(q)\B_m(q)=\sum_{t=0}^{m+n}\left[\sum_{k=0}^{\lrf{t/2}}a_k(m,n,t)f_k(q)f_{t-k}(q)\right].$$

Let $A(m,n,t)=\sum_{k=0}^{\lrf{t/2}}a_k(m,n,t)$. Then the
condition (C1) is equivalent to $A(m,n,t)\ge 0$ for $0\le t\le m+n$.
Since $\{f_k(q)\}_{k\ge 0}$ is strongly $q$-log-convex, we have
$f_0(q)f_t(q)\ge_q f_1(q)f_{t-1}(q)\ge_q
f_2(q)f_{t-2}(q)\ge_q\cdots$. By (C2) we find that
$$\sum_{k=0}^{\lrf{t/2}}a_k(m,n,t)f_k(q)f_{t-k}(q)\ge_q
\sum_{k=0}^{\lrf{t/2}}a_k(m,n,t)f_r(q)f_{t-r}(q)=A(m,n,t)f_r(q)f_{t-r}(q).$$
Thus $\{\B_n(q)\}_{n\ge 0}$
is strongly $q$-log-convex.
\end{proof}
%%%%%%%%%%%%%%%%%%%%
In what follows we will give some applications of Theorem
\ref{thm+s-q-general}.
\begin{prop}\label{prop+Morgan-Voyce}
If $\{f_k(q)\}_{k\ge 0}$ is strongly $q$-log-convex, then the
polynomials $g_n(q)=\sum_{k=0}^n\binom{n+k}{n-k}f_k(q)$ form a
strongly $q$-log-convex sequence.
\end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $a(n,k)=\binom{n+k}{n-k}$ for $0\le k\le n$. Then by
Theorem~\ref{thm+s-q-general}, it suffices to show that the triangle
$\{a(n,k)\}$ satisfies the conditions (C1) and (C2).

Let $\A_n(q)=\sum_{k=0}^n\binom{n+k}{n-k}q^k$, which is the $n$-th
Morgan-Voyce polynomial (\cite{Swa68}).  By the recurrence relation
of the binomial coefficients, we can obtain
$$\binom{n+1+k}{n+1-k}=2\binom{n+k}{n-k}+\binom{n+k-1}{n-k+1}-\binom{n-1+k}{n-1-k}.$$
From this it follows that $\A_{n+1}(q)=(2+q)\A_n(q)-\A_{n-1}(q)$ with
the initial conditions $\A_0(q)=1,\A_1(q)=1+q$ and $\A_2(q)=1+3q+q^2$.
Thus, by induction on $k$, it is easy to show
$\A_{k+1}(q)\geq_q\A_{k}(q)$ for $k\geq0$. This implies that
$$\A_{k+2}(q)-(1+q)\A_{k+1}(q)=\A_{k+1}(q)-\A_{k}(q)\geq_q0.$$
Let $m=n+k\geq n$. Then we have
\begin{eqnarray*}
&&\A_{n-1}(q)\A_{m+1}(q)-\A_n(q)\A_{m}(q)\\
&=&\A_{n-1}(q)\A_{n+k+1}(q)-\A_n(q)\A_{n+k}(q)\\
&=&\A_{n-1}(q)[(2+q)\A_{n+k}(q)-\A_{n+k-1}(q)]-[(2+q)\A_{n-1}(q)-\A_{n-2}(q)]\A_{n+k}(q)\\
&=&\A_{n-2}(q)\A_{n+k}(q)-\A_{n-1}(q)\A_{n-1+k}(q)\\
&\vdots&\\
&=&\A_{0}(q)\A_{2+k}(q)-\A_{1}(q)\A_{1+k}(q)\\
&=&\A_{k+1}(q)-\A_{k}(q) \\
&\geq_q&0.
\end{eqnarray*}
Thus the sequence $\{\A_n(q)\}_{n\ge 0}$ is strongly $q$-log-convex,
and so the condition (C1) is satisfied.


In what follows we consider (C2) condition.
Note that $a(n,k)=\binom{n+k}{n-k}=\binom{n+k}{2k}$. Let $m\geq n$, $0\leq t\leq m+n$ and $0\leq k\leq t/2$.\\
If $t$ is even and $k=t/2$, then
$$a_{k}(m,n,t)=\binom{n-1+k}{2k}\binom{m+1+k}{2k}-\binom{n+k}{2k}\binom{m+k}{2k}<0.$$
If $0\leq k<t/2$, then
\begin{eqnarray*}
a_{k}(m,n,t)&=&\left\{\binom{n-1+k}{2k}\binom{m+1+t-k}{2(t-k)}-\binom{n+k}{2k}\binom{m+t-k}{2(t-k)}\right\}\\
&+&\left\{\binom{m+1+k}{2k}\binom{n-1+t-k}{2(t-k)}-\binom{m+k}{2k}\binom{n+t-k}{2(t-k)}\right\}\\
&=&\underbrace{\frac{2[nt-(m+n+1)k]}{(n+k)[m+1-(t-k)]}\binom{n+k}{2k}\binom{m+t-k}{2(t-k)}}_{A}\\
&+&\underbrace{\frac{2[(m+n+1)k-(m+1)t]}{[n+(t-k)](m+1-k)}\binom{m+k}{2k}\binom{n+t-k}{2(t-k)}}_{B}.
\end{eqnarray*}
It can be seen that if $0\leq k<\frac{tn}{m+n+1}$, then $A>0$ and
$B<0$; if $\frac{tn}{m+n+1}<k<t/2<\frac{t(m+1)}{m+n+1}$, then $A<0$
and $B<0$. Thus $a_{k}(m,n,t)<0$ if $\frac{tn}{m+n+1}<k<t/2$. And
\begin{eqnarray*}
a_{0}(m,n,t)&=&\binom{m+1+t}{2t}-\binom{m+t}{2t}+\binom{n-1+t}{2t}-\binom{n+t}{2t}\\
&=&\frac{2t}{m+1-t}\binom{m+t}{2t}-\frac{2t}{n+t}\binom{n+t}{2t}>0
\end{eqnarray*}
Note that if $0\leq k<\frac{tn}{m+n+1}$, then $A>0$ and $B<0$. In
order to show that $A+B$ changes sign at most once (from nonnegative
to nonpositive) for $k\in[0,\frac{tn}{m+n+1})$, we consider the
monotonicity of $A/(-B)$.  Let $\Delta=A/(-B)$. Then we have
\begin{eqnarray*}
\Delta&=&\frac{[nt-(m+n+1)k][n+(t-k)](m+1-k)}{[(m+1)t-(m+n+1)k](n+k)[m+1-(t-k)]}
\times\frac{\binom{n+k}{2k}\binom{m+t-k}{2(t-k)}}{\binom{m+k}{2k}\binom{n+t-k}{2(t-k)}}\\
&=&\frac{[nt-(m+n+1)k][n+(t-k)](m+1-k)}{[(m+1)t-(m+n+1)k](n+k)[m+1-(t-k)]}
\times\frac{\binom{m+(t-k)}{m-n}\binom{m-k}{m-n}}{\binom{m+k}{m-n}\binom{m-(t-k)}{m-n}}.
\end{eqnarray*}

\begin{cl}$\Delta$ is decreasing when $k$ is
increasing.
\end{cl}
\begin{proof}
If we assume that
$$Y_k=\frac{\binom{m+(t-k)}{m-n}\binom{m-k}{m-n}}{\binom{m+k}{m-n}\binom{m-(t-k)}{m-n}},$$
then it is not hard to prove that
\begin{eqnarray*}
\frac{Y_{k+1}}{Y_k}&=&\frac{(n+t-k)(n-k)(n+1-t+k)(n+1+k)}{(m+t-k)(m-k)(m+1-t+k)(m+1+k)}\leq1,
\end{eqnarray*}
which implies that $Y_k$ is decreasing in $k$. On the other hand, it
is easy to see that both $\frac{m+1-k}{m+1-(t-k)}$ and
$\frac{n+(t-k)}{n+k}$ are decreasing in $k$. In addition,
$$\frac{nt-(m+n+1)k}{(m+1)t-(m+n+1)k}=1-\frac{(m-n+1)t}{(m+1)t-(m+n+1)k}$$
is also decreasing in $k$. Thus $\Delta$ is decreasing when $k$
increasing.
\end{proof}
By Claim 2.3, it follows that $\frac{A}{-B}-1$ changes sign at most
once (from nonnegative to nonpositive) for $k\in[0,t/2]$. So does
$A+B$. It follows that the triangle $a(n,k)=\binom{n+k}{n-k}$
satisfies the condition (C2). The proof is complete.
\end{proof}
In \cite{BSS93}, Bonin, Shapiro and Simion introduced a $q$-analog
of the large Schr\"oder number $r_n$, called the $q$-Schr\"oder
number $r_n(q)$. It is defined as:
$$r_n(q)=\sum_{P}q^{\mathrm{diag}(P)},$$
where $P$ takes over all Schr\"oder paths from $(0,0)$ to $(n,n)$
and $\mathrm{diag} (P)$ denotes the number of diagonal steps in the
path $P$. Obviously, the large Schr\"oder numbers $r_n=r_n(1)$. In
addition
$$r_n(q)=\sum_{k=0}^n\binom{n+k}{n-k}C_kq^{n-k},$$
where $C_k=\frac{1}{k+1}\binom{2k}{k}$. In \cite{Zhu13}, Zhu proved
the strong $q$-log-convexity of $q$-Schr\"oder numbers, which also
immediately follows from Proposition~\ref{prop+Morgan-Voyce}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% r_n(q)--q-LCX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{cor}\emph{\cite{Zhu13}}
The $q$-Schr\"oder numbers $r_n(q)$ form a strongly $q$-log-convex
sequence.
\end{cor}
%%%%%%%%%%%%%%% References
The $q$-central Delannoy numbers
$$D_n(q)=\sum_{k=0}^{n}\binom{n+k}{n-k}\binom{2k}{k}q^{n-k},$$
see Sagan~\cite{Sag98}. Liu and Wang~\cite{LW07} proved that numbers
$D_n(q)$ form a $q$-log-convex sequence. Zhu \cite{Zhu13}
demonstrated the strong $q$-log-convexity of $q$-central Delannoy
numbers, which can also been obtained from
Proposition~\ref{prop+Morgan-Voyce}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% D_n(q)--q-LCX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{cor}\emph{\cite{Zhu13}}
The $q$-central Delannoy numbers $D_n(q)$ form a strongly
$q$-log-convex sequence.
\end{cor}
The Bessel polynomials are defined by
$$B_n(q)=\sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\left(\frac{q}{2}\right)^k,$$
and they have been extensively studied. Chen, Wang and Yang
\cite{CWY11} obtained the strong $q$-log-convexity of $B_n(q)$.
Note that
$$B_n(q)=\sum_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k}k!\left(\frac{q}{2}\right)^k$$
and it is not hard to prove that the sequence
$\{\binom{2k}{k}k!\left(\frac{q}{2}\right)^k\}_{k\geq0}$ is strongly
$q$-log-convex. Thus we have the following corollary by
Proposition~\ref{prop+Morgan-Voyce}.
\begin{cor}\emph{\cite{CWY11}}
The Bessel polynomials $B_n(q)$ form a strongly $q$-log-convex
sequence.
\end{cor}
\begin{prop}\label{prop+binom}
If $\{f_k(q)\}_{k\ge 0}$ is strongly $q$-log-convex, then the
polynomials $b_n(q)=\sum_{k=0}^n\binom{n}{k}f_k(q)$ for $n\geq0$
form a strongly $q$-log-convex sequence.
\end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $a(n,k)=\binom{n}{k}$ for $0\le k\le n$. Then by
Theorem~\ref{thm+s-q-general}, it suffices to show that the triangle
$\{a(n,k)\}$ satisfies the conditions (C1) and (C2).

Let $g_n(q)=\sum_{k=0}^n\binom{n}{k}q^k=(1+q)^n$ for $n\geq0$. It
follows that
\begin{eqnarray*}
g_{n-1}(q)g_{m+1}(q)-g_n(q)g_{m}(q)=0
\end{eqnarray*}
for any $m\geq n$. Thus the sequence $\{g_n(q)\}_{n\ge 0}$ is
strongly $q$-log-convex, and so the condition (C1) is satisfied.

We proceed to demonstrating the condition (C2) as follows.
Note that $a(n,k)=\binom{n}{k}$. Let $m\geq n$, $0\leq t\leq m+n$ and $0\leq k\leq t/2$.
If $t$ is even and $k=t/2$, then
$$a_{k}(m,n,t)=\binom{n-1}{k}\binom{m+1}{k}-\binom{n}{k}\binom{m}{k}<0.$$
If $0\leq k<t/2$, then
\begin{eqnarray*}
a_{k}(m,n,t)&=&\left\{\binom{n-1}{k}\binom{m+1}{t-k}-\binom{n}{k}\binom{m}{t-k}\right\}\\
&+&\left\{\binom{m+1}{k}\binom{n-1}{t-k}-\binom{m}{k}\binom{n}{t-k}\right\}\\
&=&\underbrace{\frac{nt-(m+n+1)k}{n[m+1-(t-k)]}\binom{n}{k}\binom{m}{t-k}}_{A_{1}}\\
&+&\underbrace{\frac{(m+n+1)k-(m+1)t}{n(m+1-k)}\binom{m}{k}\binom{n}{t-k}}_{B_{1}}.
\end{eqnarray*}

We find that if $0\leq k<\frac{tn}{m+n+1}$, then $A_{1}>0$ and
$B_{1}<0$; if $\frac{tn}{m+n+1}\leq k<t/2<\frac{t(m+1)}{m+n+1}$,
then $A_{1}\leq0$ and $B_{1}<0$. Thus $a_{k}(m,n,t)<0$ if
$\frac{tn}{m+n+1}\leq k<t/2$. And
\begin{eqnarray*}
a_{0}(m,n,t)&=&\binom{m+1}{t}-\binom{m}{t}+\binom{n-1}{t}-\binom{n}{t}\\
&=&\frac{t}{m+1-t}\binom{m}{t}-\frac{t}{n}\binom{n}{t}>0.
\end{eqnarray*}
In order to show that $A_1+B_1$ changes sign at most once ( from
nonnegative to nonpositive) for $k\in[0,\frac{tn}{m+n+1})$, we
consider the monotonicity of $A_1/(-B_1)$.  Let
$\Delta_1=A_1/(-B_1)$. Then we have
\begin{eqnarray*}
\Delta_{1}&=&\frac{[nt-(m+n+1)k](m+1-k)}{[(m+1)t-(m+n+1)k][m+1-(t-k)]}
\times\frac{\binom{n}{k}\binom{m}{t-k}}{\binom{m}{k}\binom{n}{t-k}}.
\end{eqnarray*}
In the following we will prove that $\Delta_1$ is decreasing in $k$.


If we assume that
$$y_k=\frac{\binom{n}{k}\binom{m}{t-k}}{\binom{m}{k}\binom{n}{t-k}},$$
then it is not hard to prove that
\begin{eqnarray*}
\frac{y_{k+1}}{y_k}&=&\frac{(n-t+k+1)(n-k)}{(m-t+k+1)(m-k)}\leq1.
\end{eqnarray*}
It follows that $y_k$ is decreasing in $k$. On the other hand, we
have known that $\frac{m+1-k}{m+1-(t-k)}$ and
$\frac{nt-(m+n+1)k}{(m+1)t-(m+n+1)k}$ are decreasing in $k$. Thus
$\Delta_1$ is decreasing in $k$. It follows that $a_k(m,n,t)$
changes sign at most once (from nonnegative to nonpositive). As a
consequence, we know that $a(n,k)=\binom{n}{k}$ satisfies the
condition $(C2)$ of Theorem \ref{thm+s-q-general}. This completes
the proof.
\end{proof}
Let the Bell polynomial $B(n,q)=\sum_{k\geq0}^n S(n,k)q^k$, where
$S(n,k)$ is the Stirling number of the second kind. It was proved
that the polynomials $B(n,q)$ form a strongly $q$-log-convex
sequence, see Chen {\it et al.} \cite{CWY11} and Zhu
\cite{Zhu13,Zhu14}. Note that
$B(n+1,q)=\sum_{k=0}^n\binom{n}{k}B(n,q)$. Thus, by induction and
Proposition \ref{prop+binom}, we can give a new proof for the strong
$q$-log-convexity of $B(n,q)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let the polynomial
$$W_n(q)=\sum_{k\geq0}\binom{n}{k}^2q^k,$$
which is called the Narayana polynomials of type B,
see~\cite{CTWY10}. Liu and Wang~\cite{LW07} conjectured that
$\{W_n(q)\}_{n\geq0}$ is $q$-log-convex, which was proved by Chen
{\it et al.}~\cite{CTWY10} using the theory of symmetric functions.
In addition, Zhu \cite{Zhu13} proved the strong $q$-log-convexity of
$W_n(q)$. Now we can extend it to the following by Theorem
\ref{thm+s-q-general}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}\label{prop+squar}
If $\{f_k(q)\}_{k\ge 0}$ is strongly $q$-log-convex, then the
polynomials $s_n(q)=\sum_{k=0}^n\binom{n}{k}^2f_k(q)$ form a
strongly $q$-log-convex sequence.
\end{prop}
\begin{proof}
Note that the strong $q$-log-convexity of $W_n(q)$ has been proved,
see Zhu \cite{Zhu13}. So the condition $(C1)$ of Theorem
\ref{thm+s-q-general} is satisfied.


Note that $a(n,k)=\binom{n}{k}^{2}$, $m\geq n$, $0\leq t\leq m+n$ and $0\leq k\leq t/2$.\\
If $t$ is even and $k=t/2$, then
$$a_{k}(m,n,t)=\binom{n-1}{k}^{2}\binom{m+1}{k}^{2}-\binom{n}{k}^{2}\binom{m}{k}^{2}<0.$$
If $0\leq k<t/2$, then
%\begin{eqnarray*}
%a_{k}(m,n,t)&=&\left\{\binom{n-1}{k}^{2}\binom{m+1}{t-k}^{2}-\binom{n}{k}^{2}\binom{m}{t-k}^{2}\right\}\\
%&+&\left\{\binom{m+1}{k}^{2}\binom{n-1}{t-k}^{2}-\binom{m}{k}^{2}\binom{n}{t-k}^{2}\right\}\\
%&=&\underbrace{\frac{nt-(m+n+1)k}{n[m+1-(t-k)]}\frac{n(2m+2-t)-(m-n+1)k}{n[m+1-(t-k)]}\binom{n}{k}^{2}\binom{m}{t-k}^{2}}_{A_{2}}\\
%&+&\underbrace{\frac{(m+n+1)k-(m+1)t}{n(m+1-k)}\frac{(m+1)(2n-t)+(m-n+1)k}{n(m+1-k)}\binom{m}{k}^{2}\binom{n}{t-k}^{2}}_{B_{2}}.
%\end{eqnarray*}
\begin{align*}
a_{k}(m,n,t)=&\left\{\binom{n-1}{k}^{2}\binom{m+1}{t-k}^{2}-\binom{n}{k}^{2}\binom{m}{t-k}^{2}\right\}\\
+&\left\{\binom{m+1}{k}^{2}\binom{n-1}{t-k}^{2}-\binom{m}{k}^{2}\binom{n}{t-k}^{2}\right\}\\
=&\underbrace{\frac{nt-(m+n+1)k}{n[m+1-(t-k)]}\frac{n(2m+2-t)-(m-n+1)k}{n[m+1-(t-k)]}\binom{n}{k}^{2}\binom{m}{t-k}^{2}}_{A_{2}}\\
+&\underbrace{\frac{(m+n+1)k-(m+1)t}{n(m+1-k)}\frac{(m+1)(2n-t)+(m-n+1)k}{n(m+1-k)}\binom{m}{k}^{2}\binom{n}{t-k}^{2}}_{B_{2}}.
\end{align*}
We find that if $0\leq k<\frac{tn}{m+n+1}$, then $A_{2}>0$ and
$B_{2}<0$; if $\frac{tn}{m+n+1}\leq k<t/2<\frac{t(m+1)}{m+n+1}$,
then $A_{2}\leq 0$ and $B_{2}<0$. Thus $a_{k}(m,n,t)<0$ if
$\frac{tn}{m+n+1}\leq k<t/2$. And
\begin{eqnarray*}
a_{0}(m,n,t)&=&\binom{m+1}{t}^{2}-\binom{m}{t}^{2}+\binom{n-1}{t}^{2}-\binom{n}{t}^{2}\\
&=&\frac{t}{m+1-t}\binom{m}{t}\left\{\binom{m+1}{t}+\binom{m}{t}\right\}-\frac{t}{n}\binom{n}{t}\left\{\binom{n-1}{t}+\binom{n}{t}\right\}>0.
\end{eqnarray*}
Note that if $0\leq k<\frac{tn}{m+n+1}$, then $A_{2}>0$ and
$B_{2}<0$. Let $\Delta_{2}=A_{2}/(-B_{2})$. Then we have
$$
\Delta_{2}=y_k\times\Delta_{1}\times\frac{n(2m+2-t)-(m-n+1)k}{(m+1)(2n-t)+(m-n+1)k}\times\frac{m+1-k}{m+1-(t-k)}
$$

Noticing that when $k$ is increasing, we have known that $y_k$,
$\Delta_{1}$ and $\frac{m+1-k}{m+1-(t-k)}$ are decreasing,
respectively. Moreover, it is easy to see that
$\frac{n(2m+2-t)-(m-n+1)k}{(m+1)(2n-t)+(m-n+1)k}$ is decreasing when
$k$ increasing. So is $\Delta_{2}$. As a result, we obtain that
$a(n,k)=\binom{n}{k}^2$ satisfies the condition $(C2)$ of Theorem
\ref{thm+s-q-general}. This completes the proof.
\end{proof}


In \cite{DJM07}, Drivera, {\it et al.} proved that a conjecture of
C. Greene and H. Wilf that all zeros of the hypergeometric
polynomial
$$P_n(q)=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}q^k$$
are real. It is known that $P_n(1)$ is related to the Franel
numbers, and Sun \cite{S13} studied the congruence properties of
$P_n(q)$. In \cite{DR132}, it was proved that polynomials $P_n(q)$
 form a $q$-log-convex sequence. By Proposition
\ref{prop+squar}, we have the following stronger result.
\begin{cor}
The hypergeometric polynomials $P_n(q)$
 form a strongly
$q$-log-convex sequence.
\end{cor}
Note that the rook polynomial of a square of side $n$, denoted by
$S_n(q)$, is given by
$$S_n(q)=\sum_{k=0}^n\binom{n}{k}^2k!q^k,$$
see \cite[Chapter 3. Problems 18]{Ri} for instance, which also has
only real zeros in term of the rook theory. The following result is
immediate from Proposition \ref{prop+squar}.
\begin{cor}
The rook polynomials $S_n(q)$ form a strongly $q$-log-convex
sequence.
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Narayana number $N(n,k)$ is defined as the number of Dyck paths
of length $2n$ with exactly $k$ peaks (a peak of a path is a place
at which the step $(1,1)$ is directly followed by the step
$(1,-1)$). The Narayana numbers have an explicit expression
$N(n,k)=\frac{1}{n}\binom{n}{k}\binom{n}{k-1}$. Liu and Wang
\cite{LW07} conjectured that the Narayana polynomials
$N_n(q)=\sum_{k=0}^{n}N(n,k)q^k$ are $q$-log-convex. Using the
technique of the symmetric functions, Chen, et al. \cite{CTWY10}
proved the strong $q$-log-convexity of $N_n(q)$. Recently, Zhu
\cite{Zhu13} gave a simple proof based on certain recurrence
relation. Now, we can extend it to the next result, whose proof is
omitted for brevity.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}
If $\{f_k(q)\}_{k\ge 0}$ is strongly $q$-log-convex, then the
polynomials $\mathcal {N}_n(q)=\sum_{k=0}^nN(n,k)f_k(q)$ form a
strongly $q$-log-convex sequence.
\end{prop}

%%%%%%%%%%%%%%% References
\begin{thebibliography}{99}
%\bibitem{AE79}
%J.C. Ahuja, E.A. Enneking, Concavity property and a recurrence
%relation for associated Lah numbers, Fibonacci Quart. 17 (1979)
%158--161.
%\bibitem{Aig98}
%M. Aigner, Motzkin numbers, European J. Combin. 19 (1998) 663--675.

%\bibitem{ABHPS08}
%F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes
%and growth series of root lattices, arXiv:Math.CO/0809.5123v1.
%\bibitem{ASW52}
%M. Aissen, I.J. Schoenberg and A.M. Whitney, On the generating
%functions of totally positive sequences I, J. Analyse Math. 2 (1952)
%93--103.
%\bibitem{AKK00}
%N. Asai, I. Kubo and H.-H. Kuo, Bell numbers, log-concavity, and
%log-convexity, Acta Appl. Math. 63 (2000) 79--87.
%\bibitem{Ben96} M. Benoumhani, On Whitney numbers of Dowling
%lattices, Discrete Math. 159 (1996) 13--33.
%\bibitem{Ben97}
%M. Benoumhani, On some numbers related to Whitney numbers of Dowling
%lattices, Adv. in Appl. Math. 19 (1997) 106--116.
%\bibitem{Ben99}
%M. Benoumhani, Log-concavity of Whitney numbers of Dowling lattices,
%Adv. in Appl. Math. 22 (1999) 181--189.
%\bibitem{BB05}
%A. Bj\"orner, F. Brenti, Combinatorics of Coxeter Groups, Graduate
%Texts in Mathematics 231, Springer-Verlag, 2005.
\bibitem{BSS93}
J. Bonin, L. Shapiro, R. Simion, Some $q$-analogues of the
Schr\"oder numbers arising from combinatorial statistics on lattice
paths, J. Statist. Plann. Inference 34 (1993) 35--55.
%\bibitem{Bona12}
%M. B\'ona, Combinatorics of Permutations, second edition, CRC Press,
%Boca Raton, FL, 2012.

%\bibitem{Bon02}
%M. B\'ona, Symmetry and Unimodality in Stack sortable permutations,
%J. Combin. Theory Ser. A 98 (2002) 201--209; Corrigendum: J. Combin.
%Theory Ser. A 99 (2002) 191--194.
%\bibitem{BA00}
%M. B\'ona and R. Ehrenborg, A combinatorial proof of the
%log-concavity of the numbers of permutations with $k$ runs, J.
%Combin. Theory Ser. A 90 (2000) 293--303.
%\bibitem{Bon04}
%M. B\'ona, Combinatorics of permutations, Chapman \& Hall/CRC, Boca
%Raton, FL, 2004.
\bibitem{Bra06}
P. Br\"and\'en, On linear transformations preserving the P\'olya
frequency property, Trans. Amer. Math. Soc. 358 (2006) 3697--3716.
%\bibitem{Branden09}
%P. Br\"{a}dr\'{e}n, Iterated sequeces and the geometry of zeros, J.
%Reine Angew. Math, 658 (2011) 115--131.
%\bibitem{Bre89}
%F. Brenti, Unimodal, log-concave, and P\'olya frequency sequences in
%combinatorics,  Mem. Amer. Math. Soc. 413 (1989).
\bibitem{Bre94}
F. Brenti, Log-concave and unimodal sequences in algebra,
combinatorics, and geometry: an update, Contemp. Math. 178 (1994)
71--89.
%\bibitem{Bre94EuJC}
%F. Brenti, $q$-Eulerian polynomials arising from Coxeter groups,
%European J. Combin. 15 (1994) 417--441.
%\bibitem{Bre95}
%F. Brenti, Combinatorics and total positivity, J. Combin. Theory
%Ser. A 71 (1995) 175--218.

%\bibitem{Bre96}
%F. Brenti, The applications of total positivity to combinatorics,
%and conversely, Math. Appl. 359 (1996) 451--473.
%\bibitem{Bre00}
%F. Brenti, A class of $q$-symmetric functions arising from plethysm,
%J. Combin. Theory Ser. A 91 (2000) 137--170.
\bibitem{But90}
L.M. Butler, The $q$-log concavity of $q$-binomial coeffcients, J.
Combin. Theory Ser. A 54 (1990) 54--63.
\bibitem{BF07}
L.M. Butler, W.P. Flanigan, A note on log-convexity of $q$-Catalan
numbers. Ann. Comb. 11 (2007) 369--373.
%\bibitem{Cal00}
%D. Callan,
%Notes on Motzkin and Schr\"oder numbers, available at\\
%\verb|http://www.stat.wisc.edu/~callan/papersother/|
%\bibitem{Car71}
%L. Carlitz, Note on the numbers of Jordan and Ward, Duke Math. 38
%(1971) 783--790.
\bibitem{CTWY10}
W.Y.C. Chen, R.L. Tang, L.X.W. Wang, A.L.B. Yang, The
$q$-log-convexity of the Narayana polynomials of type B, Adv. in Appl.
Math. 44(2) (2010) 85--110.

\bibitem{CWY10}
W.Y.C. Chen, L.X.W. Wang, A.L.B. Yang, Schur positivity and the
$q$-log-convexity of the Narayana polynomials, J. Algebraic Combin.
32(3) (2010) 303--338.

\bibitem{CWY11}
W.Y.C. Chen, L.X.W. Wang, A.L.B. Yang, Recurrence relations for
strongly $q$-log-convex polynomials, Canad. Math. Bull. 54 (2011)
217--229.
\bibitem{DP49}
H. Davenport and G. P\'olya, On the product of two power series,
Canadian J. Math. 1 (1949) 1--5.


\bibitem{DR13}
D.Q.J. Dou, A.X.Y. Ren, On the $q$-log-convexity conjecture of Sun,
\arxiv{1308.2736}.

\bibitem{DR132}
D.Q.J. Dou, A.X.Y. Ren, The $q$-log-convexity of Domb's polynomials, \arxiv{1308.2961}.
\bibitem{DJM07}
K. Drivera, K. Jordaanb, A. Mart\'{i}nez-Finkelshtein, P\'{o}lya
frequency sequences and real zeros of some $_3F_2$ polynomials, J.
Math. Anal. Appl. 332 (2007) 1045--1055.

%\bibitem{DP49}
%H. Davenport and G. P\'olya, On the product of two power series,
%Canadian J. Math. 1 (1949) 1--5.

%\bibitem{DV03}
%T. Do\v sli\'c and D. Veljan, Calculus proofs of some combinatorial
%inequalities, Math. Inequal. Appl. 6 (2003) 197--209.
%\bibitem{DSV04}
%T. Do\v sli\'c, D. Svrtan and D. Veljan, Enumerative aspects of
%secondary structures, Discrete Math. 285 (2004) 67--82.

%\bibitem{Dos06}
%T. Do\v sli\'c, Logarithmic behavior of some combinatorial
%sequences, Discrete Math. 308 (2008) 2182--2212.
%\bibitem{Dow73}
%T. A. Dowling, A class of geometric lattices based on finite groups,
%J. Combin. Theory Ser. B 14 (1973) 61--86; Erratum, J. Combin.
%Theory Ser. B 15 (1973) 211.
%\bibitem{E94}
%K. Engel, On the average rank of an element in a filter of the
%partition lattice, J. Combin. Theory Ser. A 65 (1994) 67--78.

%\bibitem{Fisk08}
%S. Fisk, Polynomials, roots, and interlacing, arXiv:math/0612833.
%\bibitem{FS70}
%D. Foata, M. Sch\"utzenberger, Th\'eorie G\'eom\'etrique des
%Polyn\^omes Euleriens, Lecture Notes in Mathematics, No. 138,
%Springer-Verlag, Berlin/New York, 1970.
%\bibitem{G59}
%F.R. Gantmacher, The Theory of Matrices, vol. II, Chelsea Publ., New
%York, 1959.

%\bibitem{HLP52}
%G.H. Hardy, J.E. Littlewood, G. P\'olya, Inequalities, Cambridge
%University Press, Cambridge, 1952.

%\bibitem{Har67}
%L.H. Harper, Stirling behavior is asymptotically normal, Ann. Math.
%Statist. 38 (1967) 410--414.

%\bibitem{Kar68}
%S. Karlin, Total Positivity, Vol. I, Stanford University Press,
%Stanford, 1968.

%\bibitem{Kur72}
%D.C. Kurtz, A note on concavity properties of triangular arrays of
%numbers, J. Combin. Theory Ser. A 13 (1972) 135--139.
%\bibitem{Kur73}
%D.E. Knuth, The Art of Computer Programming, vol. 3, Fundamental
%Algorithms, Addison-Wesley, Reading, MA, 1973.

\bibitem{Le90}
P. Leroux, Reduced matrices and $q$-log-concavity properties of
$q$-Stirling numbers, J. Combin. Theory Ser. A 54 (1990) 64--84.
%\bibitem{LW-RZP}
%L.L. Liu, Y. Wang, A unified approach to polynomial sequences with
%only real zeros, Adv. in Appl. Math. 38(4) (2007) 542--560.
\bibitem{LZ14}
Z. Lin, J. Zeng, Positivity properties of Jacobi-Stirling numbers
and generalized Ramanujan polynomials, Adv. in Appl. Math. 53 (2014)
12--27.
\bibitem{LW07}
L.L. Liu, Y. Wang, On the log-convexity of combinatorial sequences,
Adv. in. Appl. Math. 39 (2007) 453--476.

%\bibitem{PS64}
%G. P\'{o}lya, G. Szeg\"o, Problems and Theorems in Analysis, Vol.
%II, 3rd ed., 1964.
%\bibitem{Ma13}
%S.M. Ma, Enumeration of permutations by number of alternating runs,
%Discrete Math. 313 (2013) 1816--1822.
%\bibitem{Ma121}
%S.M. Ma, Derivative polynomials and enumeration of permutations by
%number of interior and left peaks, Discrete Math. 312 (2012)
%405--412.

\bibitem{MW08}
S.M. Ma, Y. Wang, $q$-Eulerian polynomials and polynomials with only
real zeros. Electron. J. Combin. 15 (2008), no. 1, Research Paper
17.

%\bibitem{Monge12}
%P. Mongelli, Total positivity properties of Jacobi--Stirling
%numbers, Adv. in. Appl. Math. 48 (2012) 354--364.


%\bibitem{Pit97}
%J. Pitman, Probabilistic bounds on the coefficients of polynomials
%with only real zeroes, J. Combin. Theory Ser. A 77 (1997) 279--303.
%\bibitem{Pos81}
%M.M. Postnikov, Stable Polynomials, Nauka, Moscow, 1981 (in
%Russian).
%\bibitem{Sag88}
%B.E. Sagan, Inductive and injective proofs of log concavity results,
%Discrete Math. 68 (1988) 281--292.
\bibitem{Ri}
J. Riordan, Combinatorial Identities, New York, 1979.
%\bibitem{Sloane}
%N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences,
%http://oeis.org.
\bibitem{Sag98}
B.E. Sagan, Unimodality and the reflection principle, Ars Combin. 48
(1998) 65--72.
\bibitem{Sag92}
B.E. Sagan, Inductive proofs of $q$-log concavity, Discrete Math. 99
(1992) 289--306.
\bibitem{Sag921}
B.E. Sagan, Log concave sequences of symmetric functions and analogs
of the Jacobi-Trudi determinants, Trans. Amer. Math. Soc. 329 (1992)
795--811.


%\bibitem{Sta08}
%R.P. Stanley, Longest alternating subsequences of permutations,
%Michigan Math. J. 57 (2008) 675--687.
\bibitem{Sta89}
R.P. Stanley, Log-concave and unimodal sequences in algebra,
combinatorics, and geometry, Ann. New York Acad. Sci. 576 (1989)
500--534.
%\bibitem{SW08}
%X.-T. Su, Y. Wang, On the unimodality problems in Pascal triangle,
%Electron. J. Combin. 15 (2008), Research Paper 113.
\bibitem{SWY11}
X.T. Su, Y. Wang, Y.N. Yeh, Unimodality Problems of Multinomial
Coefficients and Symmetric Functions, Electron. J. Combin. 18(1)
(2011), Research Paper 73.
\bibitem{S13}
Z.-W. Sun, Congruences for Franel numbers, Adv. in. Appl. Math.
51(4) (2013) 524--535.
\bibitem{Swa68}
M.N.S. Swamy, Further properties of Morgan-Voyce polynomials,
Fibonacci Quart. 6 (1968) 167--175.
%\bibitem{S87}
%L.A. Sz\'{e}kely, Holiday numbers: sequences resembling to the
%Stirling numbers of the second kind, Acta Sci. Math. 48 (1985)
%459--467.
%\bibitem{Wag91}
%D.G. Wagner, The partition polynomials of a finite set system, J.
%Combin. Theory Ser. A 56 (1991) 138--159.
%\bibitem{Wag92}
%D.G. Wagner, Total positivity of Hadamard products, J. Math. Anal.
%Appl. 163 (1992) 459--483.
%\bibitem{Wag92}
%D.G. Wagner, Total positivity of Hadamard products, J. Math. Anal.
%Appl. 163 (1992) 459--483.

\bibitem{WY07}
Y. Wang and Y.-N. Yeh, Log-concavity and LC-positivity, J. Combin.
Theory Ser. A 114 (2007), 195--210.

%\bibitem{Chen081}
%W.Y.C. Chen, Log-concavity and q-Log-convexity Conjectures on the
%Longest Increasing Subsequences of Permutations,  arXiv:
%0806.3392v2.


\bibitem{Zhu13}
B.-X. Zhu, Log-convexity and strong $q$-log-convexity for some
triangular arrays, Adv. in. Appl. Math. 50(4) (2013) 595--606.
\bibitem{Zhu14}
B.-X. Zhu, Some positivities in certain triangular array, Proc.
Amer. Math. Soc. 142(9) (2014) 2943--2952.

\end{thebibliography}
\end{document}
