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\begin{document}
\begin{center}
\Large Continued Fractions for Rogers-Szeg\"o polynomials
\end{center}

\begin{center}
Qing-Hu Hou\\
Center for Combinatorics, LPMC\\
Nankai University, Tianjin 300071, P. R. China\\
Email: contact@eyou.com

Alain Lascoux\\
CNRS, Institut Gaspard Monge, Universit\'e de Marne-la-Vall\'ee\\
77454 Marne-la-Vall\'ee Cedex, France\\[2pt]
 and \\[2pt]
Center for Combinatorics, LPMC\\
 Nankai University, Tianjin 300071, P.R. China\\
Alain.Lascoux@univ-mlv.fr

Yan-Ping Mu\\
Center for Combinatorics, LPMC\\
Nankai University, Tianjin 300071, P. R. China\\
Email: myphb@eyou.com

\end{center}

\begin{abstract}
We evaluate different Hankel determinants of Rogers-Szeg\"o
polynomials, and deduce from it continued fraction expansions for
the generating function of RS polynomials. We also give an
explicit expression of the orthogonal polynomials associated to
moments equal to RS polynomials, and a decomposition of the Hankel
form with RS polynomials as coefficients.
\end{abstract}

\section{\normalsize Definitions and Notations}
Let $(a)_n=(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$. The {\it
$q$-binomial coefficient}, or the {\it Gauss polynomial}, is
$$
{n\brack k}  = {(q;q)_n
\over (q;q)_k (q;q)_{n-k}} =
{ (q^n-1)(q^{n-1}-1)\cdots (q^{n-k+1}-1) \over
(q^k-1) (q^{k-1}-1) \cdots (q-1)},
$$
which is equal to the number of $k$-dimensional vector subspaces
of the $n$-dimensional vector space $V_n(q)$ over the finite field
$ GF(q)$ if $q$ is a prime power.

The {\it
Rogers-Szeg\"o polynomials} are defined by
\begin{equation}\label{defRS}
h_n(t,q)=\sum\limits_{k=0}^n {n \brack k} t^k = (q)_n \sum {t^k \over (q)_k(q)_{n-k}}.
\end{equation}
They are some variants of $q$-Hermite polynomials which play an
important role in the theory of basic hypergeometric series
\cite{Andrews}. 
In fact, we have
\begin{equation*}
H_n(x|q)=e^{in\theta}h_n(e^{2i\theta},q), \qquad x=\cos\theta.
\end{equation*}


Rogers-Szeg\"o polynomials appear as 
coefficients in Fine's transformation
of hypergeometric series\cite[p.~27]{Fine}. 
New representations for them have been recently  given
by Berkovitch and Warnaar \cite{BerkWar}.
Distribution of zeros of Rogers-Szeg\"o polynomials  
are studied in \cite{DriverLubinsky}, \cite{LubinskySaff}. 


For $t=1$, Rogers-Szeg\"o polynomials
specialize to the Galois numbers of Goldman and Rota
\cite[p.~77]{Goldman} (cf. also \cite{Kac}):
$$  G_n = \sum_{k=0}^n {n\brack k}.$$

Let us  introduce some notations related to symmetric functions
(cf. \cite{Lascoux}).
 Given  two sets of indeterminates $A,B$
(we say \emph{alphabets}), the complete symmetric functions
$S^i(A-B)$, $i\in\Z$, of the formal difference $A-B$  are defined
by the following generating function (using  another variable
$z$)~:
\begin{equation}  \label{eq:DefComplete}
\sigma_z(A-B) = \sum_{i \ge 0} S^i(A-B) z^i = {\prod_{b \in B}
(1-bz) \over \prod_{a \in A} (1-az)}.
\end{equation}

More generally, for any polynomial $P=\sum_{(c,u)} c\, u$,
$c\in\C$, $u$ monomial, and any extra variable $q$, we define
$S^i(P)$ and $S^i(P/(1-q))$ by the generating functions
\begin{eqnarray}  \label{eq:DefComplete2}
 \sigma_z \Big( \sum\limits_{(c,u)} cu \Big) &=& \sum_{i}  z^i S^i(P)
          = \prod_{(c,u)} (1-zu)^{-c}  \  \\
  \sigma_z \left( \frac{P}{1-q}\right) &=& \sum_{i} z^iS^i\left(\frac{P}{1-q}\right)
  = \prod_{(c,u)}\prod_{j=0}^\infty (1-zq^j u)^{-c}
\end{eqnarray}

With these notations, a variable $x$ is identified with the
alphabet $\{ x\}$ of order $1$, and $(1-q)^{-1}$ with the infinite
alphabet $\{1,q,q^2,\ldots \}$.
 Any formal series $\sum_{n\geq 0} z^n S^n$, with $S^0=1$,
 can be formally
factorized, i.e. identified with $\sigma_z(A)$ for some alphabet
$A$ such that $S^n=S^n(A)$, $n\in\N$.

Our basic ring ${\mathcal R}$ is the ring of rational functions in
two variables $q,t$ and some alphabets $A,B,\ldots$.

For any element $\theta\in {\mathcal R}$, any $n\in\N$, any
$I=(i_1,\ldots,i_n) \in \N^n$, the \emph{Schur function}
$S_I(\theta)$ is defined by the determinant \cite{Macdonald}~:
$$
S_{I}(\theta) = \big| S^{i_s+s-r} (\theta) \big|_{1 \le r,s \le
n}.
$$

\section{\normalsize Recursion for Rogers-Szeg\"o Polynomials}

Taking coefficients of powers of $z$ in (\ref{eq:DefComplete}),
one gets
\begin{equation}
\label{expansion} S^n (A \pm B) = \sum_{k=0}^n S^k(A)\,
S^{n-k}(\pm B).
\end{equation}

{} From \eqref{expansion}, we see that the Rogers-Szeg\"o Polynomial
$h_n(t,q)$ can be written as:
$$
(q)_n S^n\left({1+t \over  1-q}\right).
$$

\begin{lem}
\label{lem-rec} Let $B,C$ be two alphabets of order less than or
equal to $m$. Then $\S_n :=S^n \left(\ {B-C \over 1-q} \right)$,
$n\in\N$, satisfies the following recurrence:
\begin{equation}
\label{eq-BC} \sum_{r=0}^m \big(S^r(-B)-S^r(-C) q^{n-r}\big)\,
\S_{n-r} = 0.
\end{equation}
\end{lem}

\pf
Writing $-B+{B-C \over 1-q} = -C + q{B-C \over 1-q}$, then from \eqref{expansion}, together
with the fact that $S^n (-B) = 0 = S^n(-C)$, for $\ n > m$, we get that
$$
\sum S^r(-B) S^{n-r}\left( {B-C \over 1-q} \right) =
  \sum S^r(-C) q^{n-r} S^{n-r}\left( {B-C \over 1-q} \right).
$$
\qed

Lemma 2.1 implies the following recursion \cite{Szego}:
\begin{coro}
Let $h_n(t,q)$ be the Rogers-Szeg\"o polynomial. Then
\begin{equation}
\label{eq-Grec}
h_n(t,q) = (1+t) h_{n-1}(t,q) - t (q^{n-1}-1) h_{n-2}(t,q), \quad 
n \ge 2.
\end{equation}
\end{coro}
\pf Taking $B=\{ 1,t \}, C=\emptyset$ in Lemma~\ref{lem-rec}, we
have
\begin{equation}
\label{eq-Srec}
(1-q^n) S^n \left( {1+t \over 1-q} \right) -(1+t) S^{n-1} \left( {1+t \over 1-q} \right)
+t S^{n-2} \left( {1+t \over 1-q} \right) =0.
\end{equation}

Substituting $S^n \left( {1+t \over 1-q} \right)=h_n(t,q)/(q)_n$
in Equation~\eqref{eq-Srec},
we derive the recursion immediately. \qed

{\noindent \it Remark.}
Goldman and Rota \cite{Goldman} proved the recursion
$$G_{n+1}=2G_n +(q^n-1) G_{n-1} \ ,\ G_0=1 \ ,\ G_1= 2 \ , $$
using vector spaces over finite fields. Nijenhuis, Solow and Wilf
\cite{Nijenhuis} gave a bijective proof for the same recursion. K.
Hikami and B. Basu-Mallick \cite{Hikami} gave also a recursion for
generalized Rogers-Szeg\"o polynomials.


\section{\normalsize Determinants of Rogers-Szeg\"o Polynomials}
\begin{theo}
\label{th-S} Let $\A$ be the alphabet defined by
$S^i(\A)=h_i(t,q)$, $i\in\N$. For $m \ge 0$, we have
\begin{eqnarray}
S_{n^n} (\A) &=& t^{n \choose 2} q^{n+1 \choose 3}
  (q)_0 (q)_1 \cdots (q)_{n-1} h_n(t,q^{-1}),\label{eq-1}\\
S_{n^n,m}(\A) &=& (q)_0 (q)_1 \cdots (q)_n \label{eq-2}
  t^{n+1 \choose 2} q^{n+1 \choose 3} {m \brack n} h_{m-n}(t,q) ,
\end{eqnarray}
where $h_n(t,q)$ are regarded as $0$ for $n<0$.
In particular,
\begin{equation}
\label{eq-3}
S_{n^{n+1}}(\A) = (q)_0 (q)_1 \cdots (q)_n
  t^{n+1 \choose 2} q^{n+1 \choose 3}.
\end{equation}
\end{theo}

\pf Denote $X={1+t \over 1-q}$. By definition,
\begin{eqnarray}
S_{n^n}(\A) &=& \big| h_{n+j-i}(t,q) \big|_{1\le i,j \le n} \nonumber \\[5pt]
&=& \big| (q)_{n+j-i} S^{n+j-i}(X) \big|_{1\le i,j \le n}
 \nonumber \\[5pt]
&=& (q)_1 \cdots (q)_n
\cdot \Big| (q^{n-i+2})_{j-1} S^{n+j-i}(X) \Big|_{1\le i,j \le n}. \label{Snn}
\end{eqnarray}

We can write recursion~\eqref{eq-Srec} as
\begin{eqnarray}
\lefteqn{(q^{k})_{m} S^{k+m-1}(X)} \nonumber \\
&=& (1+t) (q^k)_{m-1} S^{k+m-2}(X) - t (1-q^{m-1}) (q^k)_{m-2} S^{k+m-3}(X) \nonumber \\
&& \quad - t q^{m-1} (q^{k-1})_{m-1} S^{k+m-3}(X) \label{mul-r}.
\end{eqnarray}

Applying identity~\eqref{mul-r} on columns $n, n-1, \ldots, 2$ of the determinant
in \eqref{Snn} successively, it becomes
\begin{eqnarray*}
\lefteqn{(-t)^{n-1} q^{n-2} q^{n-3} \cdots q^0} \\
&& \times
\left|
\begin{array}{ccccc}
S^n(X) & S^{n-1}(X) & (q^n)_1 S^n(X) & \cdots
  & (q^n)_{n-2} S^{2n-3}(X) \\[3pt]
S^{n-1}(X) & S^{n-2}(X) & (q^{n-1})_1 S^{n-1}(X) & \cdots
  & (q^{n-1})_{n-2} S^{2n-4}(X) \\[3pt]
\cdots & \cdots & \cdots & \cdots \\[3pt]
S^2(X) & S^1(X) & (q^2)_1 S^2(X) & \cdots & (q^2)_{n-1} S^{n-1}(X) \\
S^1(X) & S^0(X) & (q)_1 S^1(X) & \cdots & (q)_{n-2} S^{n-2}(X) \\
\end{array}
\right|.
\end{eqnarray*}
Repeating the same transformation, we will finally reach
\begin{eqnarray*}
&&(-t)^{n \choose 2} q^{n \choose 3}
\left|
\begin{array}{cccc}
S^n(X) & S^{n-1}(X) &  \cdots
  & S^1(X) \\[3pt]
S^{n-1}(X) & S^{n-2}(X) &  \cdots
  & S^0(X) \\[3pt]
\cdots & \cdots & \cdots \\[3pt]
S^2(X) & S^1(X) & \cdots & 0 \\
S^1(X) & S^0(X) & \cdots & 0 \\
\end{array}
\right| = t^{n \choose 2} q^{n \choose 3} S_{1^n}(X).
\end{eqnarray*}

Noting that
$$
S_{1^n}(X) = (-1)^n S^n \left( -{1+t \over 1-q} \right)
= {q^{n \choose 2} \over (q)_n} h_n(t,q^{-1}),
$$
we obtain
$$
S_{n^n} (\A) = t^{n \choose 2} q^{n+1 \choose 3} (q)_1 \cdots
(q)_{n-1} h_n(t,q^{-1}).
$$

For $m \ge n$, define
\begin{eqnarray*}
\lefteqn{W_{n,m} ={S_{n^n,m}(\A) \over (q)_0 (q)_1 \cdots (q)_n}}\\[5pt]
&=&
{\small
\left|
\begin{array}{ccccc}
S^n(X) & (q^{n+1})_1 S^{n+1}(X) & \cdots & (q^{n+1})_{n-1} S^{2n-1}(X) & (q^{n+1})_m S^{n+m}(X) \\[3pt]
S^{n-1}(X) & (q^n)_1 S^n(X) & \cdots & (q^n)_{n-1} S^{2n-2}(X) & (q^n)_m S^{n+m-1}(X) \\[3pt]
\cdots & \cdots & \cdots & \cdots & \cdots \\[3pt]
S^1(X) & (q^2)_1 S^2(X) & \cdots & (q^2)_{n-1} S^n(X) & (q^2)_m S^{m+1}(X)\\
S^0(X) & (q)_1 S^1(X) & \cdots & (q)_{n-1} S^{n-1}(X) & (q)_m S^m(X)
\end{array}
\right|
}.
\end{eqnarray*}

Applying identity~\eqref{mul-r} on the last column of $W_{n,m}$, we transform it into
\begin{eqnarray*}
\lefteqn{W_{n,m} = (1+t)W_{n,m-1} - t(1-q^{m-1}) W_{n,m-2}} \\[5pt]
&& + \
\left|
\begin{array}{cccc}
S^n(X) &  \cdots & (q^{n+1})_{n-1} S^{2n-1}(X) & (q^n)_{m-1} S^{n+m-2}(X) \\[3pt]
S^{n-1}(X) & \cdots & (q^n)_{n-1} S^{2n-2}(X) & (q^{n-1})_{m-1} S^{n+m-3}(X) \\[3pt]
\cdots & \cdots & \cdots & \cdots \\[3pt]
S^1(X) & \cdots & (q^2)_{n-1} S^n(X) & (q)_{m-1} S^{m-1}(X)\\
S^0(X) & \cdots & (q)_{n-1} S^{n-1}(X) & 0
\end{array}
\right|
\end{eqnarray*}
After successive such transformations, one finally obtains
\begin{equation}
\label{w-r}
W_{n,m}=(1+t)W_{n,m-1}-t(1-q^{m-1})W_{n,m-2}+t^n q^{m-1} q^{n-1 \choose 2} W_{n-1,m-1}.
\end{equation}
Noting that
$$
t^{n+1 \choose 2} q^{n+1 \choose 3} {m \brack n} h_{m-n}(t,q)
$$
satisfies the recursion \eqref{w-r} and coincides with $W_{n,m}$ for $(0,m)$, $(n,0)$ and
$(n,1)$, we get \eqref{eq-2}. \qed

\section{\normalsize Continued Fractions for Rogers-Szeg\"o Polynomial}
There are three main types of continued fraction expressions for
formal series. They are all obtained by evaluating some Hankel
determinants, 
i.e. determinants which are constant along anti-diagonals. 
What we write $S_{k^n}(\A)$ would usually appear as
$$ \begin{vmatrix}\,  c_{k-n+1} & \cdots & c_k \\
                      \vdots &       &\vdots \\
                     c_k &   \cdots       & c_{k+n-1}
 \, \end{vmatrix}  
\ \longleftrightarrow  \ 
\pm \begin{vmatrix}  S_k(\A) &\cdots & S_{k+n-1}(\A)\, \\
                     \vdots  &       & \vdots \\
                   \, S_{k-n+1}(\A) & \cdots & S_k(\A) 
   \end{vmatrix}  \ ,
    $$ 
the $c_i= S_i(\A)$ being the moments.     

We refer to Stieltjes \cite{Stieltjes}, Wall
\cite{Wall} and Brezinski \cite{Brezinski}. The determinants given
by Theorem~\ref{th-S} allow us to write the following three
continued fractions. We do not treat questions of convergence,
but only expand formally continued fractions.


\begin{theo}
\label{GF}
The generating function of Rogers-Szeg\"o polynomials $h_n(t,q)$ 
is equal to each of
the three following continued fractions:
\begin{eqnarray*}
\sum_{n=0}^\infty h_n(t,q) z^n &=&
 {1 \over {\displaystyle 1 - {\displaystyle z \g_1 \over {
    \displaystyle 1 - {\displaystyle z \g_2 \over
    {\displaystyle 1 - {\displaystyle z \g_3 \over \cdots}}}}}}} \\
&=& {1 \over {\displaystyle 1-\z_0 z + {\displaystyle \b_1 z^2 \over {
    \displaystyle 1 - \z_1 z + {\displaystyle \b_2 z^2 \over
    {\displaystyle 1- \z_2 z + {\displaystyle \b_3 z^2 \over \cdots}}}}}}}, \\[5pt]
\sum_{n=0}^\infty h_n(t,q) z^{-n-1}
 &=&
{\displaystyle \frac {1}{z + {\displaystyle \frac {1}{ -
{\displaystyle \frac {1}{\a_1}}  + {\displaystyle \frac {1
}{{\displaystyle \frac {z}{\a_2}}  + {\displaystyle
\frac {1}{ - {\displaystyle \frac {1}{\a_3} + \cdots }
 }} }} }} }},
\end{eqnarray*}
where
\begin{eqnarray*}
\a_m & = &
\Bigg\{
\begin{array}{ll}
\displaystyle
(-1)^{n-1}
{q^{n \choose 2} h_n(t,q^{-1})
h_{n-1}(t,q^{-1}) \over t^{n-1} (q)_{n-1} }, & \quad m=2n-1, \\[5pt]
\displaystyle
(-1)^n {t^n (q)_n \over q^{n \choose 2} h^2_n(t,q^{-1})},  &  \quad m=2n,
\end{array}  \\[10pt]
\g_m &=& \Bigg\{
\begin{array}{ll}
\displaystyle
{ q^{n-1}  h_n(t,q^{-1}) \over h_{n-1}(t,q^{-1})}, & \quad m=2n-1, \\[10pt]
\displaystyle
- t (1-q^n) {h_{n-1}(t,q^{-1}) \over h_{n}(t,q^{-1})}, & \quad m=2n,
\end{array} \\[10pt]
\b_n &=& q^{n-1} t (1-q^n), \\[5pt]
\z_n &=& q^n (1+t).
\end{eqnarray*}
\end{theo}

\noindent {\it Proof.}
In our notations, Stieltjes' parameters, for the series $\sum z^n S^n(\A)$, are
\cite{Lascoux}
\begin{eqnarray*}
\a_m & = &
\Bigg\{
\begin{array}{ll}
\displaystyle
(-1)^{n-1} \ {S_{n^n}(\A) S_{(n-1)^{n-1}}(\A) \over S_{(n-1)^n}(\A)^2}, & \quad m=2n-1, \\[5pt]
\displaystyle (-1)^n \ {S_{n^{n+1}}(\A) S_{(n-1)^n}(\A) \over
S_{n^n}(\A)^2}, & \quad m=2n,
\end{array} \\
\b_n & = & {S_{n^{n+1}}(\A) S_{(n-2)^{n-1}}(\A) \over S_{(n-1)^n}(\A)^2} = - \g_{2n} \g_{2n-1},\\[5pt]
\g_n & = & \a_n \a_{n-1},\\[5pt]
\z_n & = & {S_{n^n,n+1}(\A) \over S_{n^{n+1}}(\A) } - { S_{(n-1)^{(n-1)},n}(\A) \over S_{(n-1)^n}(\A) }.
\end{eqnarray*}
\qed


\noindent {\it Remark.}
Note that the usual generating function considered for the 
Rogers-Szeg\"o polynomials is
$$
\sum_{n \ge 0} {h_n(t;q) \over (q;q)_n} z^n 
= {1 \over (z;q)_\infty (zt;q)_\infty},
$$
which is different from the one in Theorem~\ref{GF}.


\section{\normalsize Hankel Forms}

To any alphabet $\A$ is associated a Hankel form
$$
  H(\A;x) = \sum_{i,j=0}^\infty   S_{i+j}(\A)\,  x_i x_j  \ .
$$
The problem of decomposing Hankel forms into sums of squares has
also been solved by Stieltjes \cite{Baillaud}. He found the
coefficients, that we denote by $\star$, such that
$$ H(\A;x) = (x_0+ \star x_1 + \cdots)^2 +
    \star (x_1+ \star x_2 + \cdots)^2 +
   \star (x_2+ \star x_3 + \cdots)^2 + \cdots \ .$$
\begin{theo}[Stieltjes]
Let $\A$ be a generic alphabet. With $\A^0=\A$, define the alphabets
$\A^1, \A^2, \ldots$ by the generating series
$$
\sigma_z(\A^k) = S_{k^{k+1}}(\A)^{-1} \sum_{j=0}^\infty S_{k^k,\, k+j}(\A) z^j \ .
$$
Then the form $K(\A;\, x,y) = \sum\limits_{i,j=0}^\infty
S_{i+j}(\A) x^i y^j$ decomposes as
\begin{eqnarray*}  \label{eq:quadra3}
K(A;\,x,y) &=& \sigma_1 \Big( (x + y)\A^0  \Big) -
   xy S_{11}(\A)\, \sigma_1 \Big((x + y)\A^1  \Big)
       + \cdots \\
&&\quad +\cdots +\, (-xy)^n  \frac{S_{n^{n+1}}(\A)}{S_{(n-1)^n}(\A)}
       \sigma_1 \Big( (x + y)\A^n \Big)  +\cdots.
\end{eqnarray*}
\end{theo}

Substituting Equation~\eqref{eq-2} into the above identity, we instantly derive that
\begin{coro}
For Rogers-Szeg\"o polynomials $h_n(t,q)$, we have
\begin{multline*}
\sum_{i,j=0}^\infty h_{i+j} (t,q) x^i y^j
= \sum_{n=0}^\infty (-t)^n (q)_n q^{n \choose 2} \\
  \times \left( \sum_{u=0}^\infty h_u(t,q) {n+u \brack u} x^{n+u} \right)
  \left( \sum_{v=0}^\infty h_v(t,q) {n+v \brack v} y^{n+v} \right)
\end{multline*}
\end{coro}

Comparing the coefficients of $t^m s^n$ of both sides, we get the
following formula  due to Askey and Ismail \cite{Askey}:
$$
h_{m+n} (t,q) = \sum_{k=0}^{\min\{m,n\}} {n \brack k} {m \brack k}
  (-t)^k (q)_k q^{k \choose 2} h_{m-k} (t,q) h_{n-k} (t,q).
$$

\section{\normalsize Orthogonal Polynomials}
Let $\int$ be a linear functional on polynomials in $x$ such that
there exists a family of {\it orthogonal polynomials} $\{P_n(x)\}$:
$$
\int P_m(x) P_n(x)  = 0 \ {\rm if }\ m\neq  n \ , \
           \int P_n(x)P_n(x)  = c_n \not= 0\ .
$$
$c_n$ is called the normalization constant.

Identifying the {\it moments} $\int x^n$ to the complete functions
$S^n(A)$ of some alphabet $A$, we can rewrite the classical
expressions of orthogonal polynomials in term of symmetric
functions:
\begin{equation}\label{orth}
P_n(x) = S_{n^n}(A-x), \ c_n = (-1)^n S_{(n-1)^n}(A)\,  S_{n^{n+1}}(A),
 \quad  n \ge 0.
\end{equation}


In the case where we take as moments the Rogers-Szeg\"o polynomials,
the corresponding orthogonal polynomials have a simple expression
which is a corollary of combinatorial properties of Al-Salam-Chihara
polynomials obtained by 
 Ismail and Stanton\cite[Th.~6, Cor.~1]{IsmailStanton}. 
We give a direct algebraic
proof in the following theorem.


\begin{theo}
Let $\int_\A$ be the linear functional with moments equal to the
Rogers-Szeg\"o polynomials: $\int_\A x^n = h_n(t,q)$. Then
$$
P_n(x)= \sum_{k=0}^n (-1)^k {n \brack k}
q^{k \choose 2} h_{k}(t,q^{-1}) x^{n-k}
= (q)_n S^n\left({x-1-t \over 1-q}\right),
$$
are its associated orthogonal polynomials, with normalization constant
$$
c_n = t^n q^{n \choose 2} (q-1)(q^2-1)\cdots(q^n-1).
$$
\end{theo}
\pf
Start with
$$
S_{n^n}(\A-x) =
\left|
\begin{array}{ccccc}
S^n(\A) & S^{n+1}(\A) & \cdots & S^{2n-1}(\A) & x^n \\
S^{n-1}(\A) & S^n(\A) & \cdots & S^{2n-2}(\A) & x^{n-1} \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
S^1(\A) & S^2(\A) & \cdots & S^n(\A) & x^1 \\
S^0(\A) & S^1(\A) & \cdots & S^{n-1}(\A) & 1
\end{array}
\right|,
$$
and write $X={1+t \over 1-q}$. The same kind of transformation as
for Theorem~\ref{th-S} allows us to write the preceding
determinant as
\begin{eqnarray*}
\lefteqn{
(q)_0 (q)_1 \cdots (q)_n
\left|
\begin{array}{cccc}
S^n(X) & \cdots & (q^{n+1})_{n-1} S^{2n-1}(X) & x^n/(q)_n \\
S^{n-1}(X) & \cdots & (q^n)_{n-1} S^{2n-2}(X) & x^{n-1}/(q)_{n-1} \\
\cdots & \cdots & \cdots & \cdots \\
S^1(X) & \cdots & (q^2)_{n-1} S^n(X) & x^1/(q)_1 \\
S^0(X) & \cdots & (q)_{n-1} S^{n-1}(X) & 1
\end{array}
\right|}\\[5pt]
&=&
(q)_0 (q)_1 \cdots (q)_n (-t)^{n \choose 2} q^{n \choose 3} \\
&& \quad
\left|
\begin{array}{ccccc}
S^n(X) & S^{n-1}(X) & \cdots & S^1(X) & x^n/(q)_n \\
S^{n-1}(X) & S^{n-2}(X) & \cdots & S^0(X) & x^{n-1}/(q)_{n-1} \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
S^1(X) & S^0(X) & \cdots & 0 & x^1/(q)_1 \\
S^0(X) & 0 & \cdots & 0 & 1
\end{array}
\right|\\[5pt]
&=&
(q)_1 \cdots (q)_n t^{n \choose 2} q^{n \choose 3}
\sum_{k=0}^n (-1)^{n-k} {x^{n-k} \over (q)_{n-k}} S_{1^k}(X) \\[5pt]
&=&
(-1)^n (q)_1 \cdots (q)_{n-1} t^{n \choose 2} q^{n \choose 3}
\sum_{k=0}^n (-1)^k {n \brack k}
q^{k \choose 2} h_{k}(t,q^{-1}) x^{n-k},
\end{eqnarray*}

After normalization, we finally get the polynomials
$$
P_n(x)= \sum_{k=0}^n (-1)^k {n \brack k}
q^{k \choose 2} h_{k}(t,q^{-1}) x^{n-k}
= (q)_n S^n\left({x-1-t \over 1-q}\right),
$$
with normalization constant
$$
c_n = t^n q^{n \choose 2} (q-1)(q^2-1)\cdots(q^n-1).
$$
\qed

{\small {\bf Acknowledgments.} This work was done under the
auspices of the National ¡°973¡± Project on Mathematical
Mechanization and the National Science Foundation of China.
We thank the referees and D. Stanton for their comments.

\begin{thebibliography}{99}
\bibitem{Andrews}
G.E.~Andrews, \emph{The Theory of Partitions}, Addison-Wesley
Publishing Co., Mass-London-Amsterdam, 1976.

\bibitem{Askey}
R.~Askey and M.E.H.~Ismail, \emph{A generalization of
ultraspherical polynomials}, In: Studies in Pure Mathematics,
P.~Erd\"{o}s, Ed., pp.~55-78, Birkh\"{a}user, Boston, MA, 1983.

\bibitem{Baillaud}
B. Baillaud, H. Bourget, \emph{Correspondance d'Hermite et de
Stieltjes}, Gauthier-Villars, Paris, 1905.


\bibitem{BerkWar} A. Berkovich and O. Warnaar,
\emph{Positivity preserving transformations for $q$-binomial coefficients},
 {\tt math.CO/0302320} (2003). 

\bibitem{Brezinski}   C. Brezinski,
\emph{History of continued fractions and Pad\'e approximants},
Springer, 1991.


\bibitem{DriverLubinsky}
K. A. Driver and D. Lubinsky, 
\emph{Convergence of Pad\'e Approximants for a
q-Hypergeometric Series}, Aequationes Math., {\bf 42} (1991), 85--106.



\bibitem{Fine}N. J. Fine, \emph{Basic hypergeometric series and
application}, Mathematical Surveys and Monographs, Vol. 27,
Amer. Math. Soc., Providence, 1988.


\bibitem{Goldman}
J.~Goldman and G.-C. Rota, \emph{The number of subspaces of a
vector space}, in: Recent Progress in Combinatorics, W. T. Tutte
ed., Academic Press, 1969, pp. 75-83.

\bibitem{Hikami}
K. Hikami and B. Basu-Mallick, \emph{Supersymmetric Polychronakos
spin chain: motifs, distribution functions and characters},
Nuclear Physics B {\bf 566} (2000) 511--528.


\bibitem{IsmailStanton} M. Ismail and D. Stanton, 
  \emph{ More orthogonal polynomials as moments},
   Mathematical Essays in Honor of Gian-Carlo-Rota, Birkhauser (1998)
    377--396.)


\bibitem{Kac}
V. Kac and P. Cheung, \emph{Quantum Calculus}, Springer-Verlag,
New York, 2002, pp. 25--26.

\bibitem{Lascoux}
A.~Lascoux, \emph{Symmetric functions and 
combinatorial operators on polynomials},
CBMS-AMS Conference Lecture Notes (to appear).


\bibitem{LubinskySaff}
D. Lubinsky and E. B. Saff, \emph{Convergence of Pad\'e Approximants of Partial
Theta Functions and the Rogers-Szego Polynomials}, Constructive
Approximation, {\bf 3} (1987), 331--361.


\bibitem{Macdonald}
I.G.~Macdonald, \emph{Symmetric functions and Hall polynomials},
Clarendon press, Oxford, 1979.

\bibitem{Nijenhuis}
A. Nijenhuis, A.E. Solow and H.S. Wilf, \emph{Bijective methods in
the theory of finite vector spaces}, J. Combin. Theory (A) {\bf
37} (1984) 80-84.

\bibitem{Szego}
G.~Szeg\"o, \emph{Ein Beitrag zur Theorie der Thetafunktionen},
Sitz.~Preuss.~Akad.~Wiss., Phys.-Math. Klasse {\bf 19} (1926)
249-252 (reprinted in \emph{Collected Papers vol 1}, R.~Askey ed.,
(Boston, MA: Birkha\"user) pp. 795--805).

\bibitem{Stieltjes} T.J. Stieltjes, \emph{Recherches sur les fractions
continues}, Ann. Fac. Sc. Toulouse {\bf 8} (1894) 1--122.

\bibitem{Wall} H.S. Wall, \emph{Analytic theory of Continued
Fractions}, Chelsea reprint, 1973.

%\Blue{ \bibitem{Wynn} P. Wynn,
% \emph{A General System of Orthogonal Polynomials}, Quart J. Math. Oxford,
% {\bf 18} (1967), 81--96.  }

\end{thebibliography}
\end{document}


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