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\title{\bf On certain formulas of Karlin and Szeg\"o}
%
\author{\rm Bernard {\sc Leclerc} \\ [2mm]
\footnotesize \it D\'epartement de Math\'ematiques,\\
\footnotesize \it Universit\'e de Caen, 14032 Caen cedex, France.}
\date{}
%\input macro.tex

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\begin{document}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  ABSTRACT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 1cm

\begin{abstract}
Some identities due to Karlin and Szeg\"o 
which provide a relationship between 
determinants of classical orthogonal polynomials 
of Wronskian and Hankel type are 
shown to be specializations of a general algebraic
identity between minors of a matrix.

\bigskip
\centerline{\bf R\'esum\'e}

\vskip 2mm
On montre que des familles d'identit\'es d\'ecouvertes
par Karlin et Szeg\"o, qui relient des Wronskiens et des 
d\'eterminants de Hankel de polyn\^omes orthogonaux classiques,
r\'esultent par sp\'ecialisation d'une identit\'e
alg\'ebrique g\'en\'erale entre mineurs d'une matrice.
\medskip

\end{abstract}

\vskip 0.6cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  SECTION 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
\label{SECT1}
%
Let
\begin{equation}
P_n(x)= {1\over 2^n}\sum_{m=0}^{\lfloor {n\over 2}\rfloor}
(-1)^m {2n-2m \choose n} {n \choose m} x^{n-2m}
\end{equation}
denote the $n$th Legendre polynomial.
It was found some fifty years ago by Tur\'an \cite{Tu,Sze1}
that for $x\in ]-1, 1[$ and all $n\ge 0$, there holds
\begin{equation}\label{TURAN}
\left|
\matrix{ P_n(x) & P_{n+1}(x) \cr\noalign{\vskip 1mm}  
         P_{n+1}(x) & P_{n+2}(x)
} \right|
< 0 \,.
\end{equation}
Tur\'an's inequality was soon generalized in several 
ways and there is today a huge literature on Tur\'an
type inequalities (see 
\cite{As1,As2,BSS,Da1,Da2,Fo,Ga1,Ga2,Il,KS,Pa,Sk,Sza1,Sza2,VLR}
and references therein).

A major contribution to this topic was made by Karlin and Szeg\"o
\cite{KS}, who showed that for even $l$, the Hankel
determinant
\begin{equation}
T(l,n;x) = 
\left|   
\matrix{ Q_n(x) & Q_{n+1}(x)&\cdots & Q_{n+l-1}(x) 
\cr\noalign{\vskip 1mm}  
Q_{n+1}(x) & Q_{n+2}(x) &\cdots & Q_{n+l}(x) 
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots
\cr\noalign{\vskip 1mm} 
Q_{n+l-1}(x) & Q_{n+l}(x)&\cdots & Q_{n+2l-2}(x)
} \right| 
\end{equation}
has a constant sign for $x\in I$ in each of the following cases:
\begin{quote}
(i) $Q_n(x) = P_n^{(\lambda)}(x)$ (ultraspherical polynomials,
which contain Legendre polynomials for $\lambda = 1/2$) and 
$I=]-1,1[$,

(ii) $Q_n(x) = L_n^{(\alpha)}(x)$ (Laguerre polynomials) and 
$I=]0,+\infty[$,

(iii) $Q_n(x) = H_n(x)$ (Hermite polynomials) and 
$I=]-\infty , +\infty[$.
\end{quote}
Their strategy was to express the determinant $T(l,n;x)$
in terms of the  Wronskian of certain orthogonal polynomials of
another class.
For instance, in the case of Legendre polynomials they proved
that
\[
\left|    
\matrix{ P_n(x) & P_{n+1}(x)&\cdots & P_{n+l-1}(x) 
\cr\noalign{\vskip 1mm}
P_{n+1}(x) & P_{n+2}(x) &\cdots & P_{n+l}(x)  
\cr\noalign{\vskip 1mm} 
\vdots & \vdots & & \vdots 
\cr\noalign{\vskip 1mm}  
P_{n+l-1}(x) & P_{n+l}(x)&\cdots & P_{n+2l-2}(x)
} \right|
=A_{l,n} (x^2-1)^{l(n+l-1)/2}\quad\quad\quad\quad\quad
\]
\begin{equation}\label{KSZ1}
\hfill\qquad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\times \left|     
\matrix{   
T_l(u) & T_{l+1}(u)&\cdots & T_{l+n-1}(u)
\cr\noalign{\vskip 1mm} 
T'_l(u) & T'_{l+1}(u)&\cdots & T'_{l+n-1}(u)
\cr\noalign{\vskip 1mm}  
\vdots & \vdots & & \vdots  
\cr\noalign{\vskip 1mm}   
T^{(n-1)}_l(u) & T^{(n-1)}_{l+1}(u)&\cdots & T^{(n-1)}_{l+n-1}(u)
} \right|\,,
\end{equation}
where $T_l(u)$ is the $l$th Tchebichev polynomial of the second
kind, 
\begin{equation}
u=-x(x^2-1)^{-1/2}\,, \quad 
T_m^{(k)}(u) = {d^kT_m(u)\over du^k}
\end{equation}
and 
\begin{equation}
A_{l,n}= {(-1)^l\over 2} \, \prod_{p=0}^{l-1} 2^{1-n-2p}
\prod_{q=0}^{n-1}{2^{1-q}\over q!} \,.
\end{equation}
In each of the three cases (i) (ii) (iii), Karlin and Szeg\"o
managed to find and prove a similar formula, which allowed them to
reduce their analysis to that of the sign of a Wronskian
determinant, which is easier.

Thus a kind of duality emerged between Hankel and Wronskian
determinants of classical orthogonal polynomials.
However the proofs of Karlin and Szeg\"o did not clearly
show what in their formulas resulted from a general algebraic
transformation, and what in contrast was due to some particular
properties of the orthogonal polynomials under consideration.
%Also no explanation was provided for the normalization of
%the polynomials in the Hankel determinant, which happens
%to be crucial for the existence of this type of formula.
%And the question remained whether some other classical family,
%like for example the more general Jacobi polynomials, might
%fit into this scheme.
%
In trying to clarify this, we obtained a different 
derivation of the identities of Karlin and Szeg\"o, which consists
of two steps:
%\begin{quote}

\bigskip
\noindent
{\it Step 1.}\ A completely general algebraic identity stating that a Wronskian
of orthogonal polynomials is proportional to a Hankel determinant
whose elements form a new sequence of polynomials.

\bigskip
\noindent
{\it Step 2.}\ The verification that in each case considered by Karlin and 
Szeg\"o, this new sequence is, after change of variable and 
normalization, another class of classical orthogonal polynomials.
%\end{quote}

\bigskip
To be more explicit,
let us consider a sequence of arbitrary numbers $a_n,\ n\ge 0$.
With this sequence are associated two classes of polynomials
in the indeterminate $u$, defined~by:
\begin{eqnarray}
p_n(u)& =& \left|
\matrix{
a_0 & a_1 & \cdots & a_{n-1} & 1  
\cr\noalign{\vskip 1mm}
a_1 & a_2 & \cdots & a_n & u
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots & \vdots
\cr\noalign{\vskip 1mm}
a_{n-1} & a_n & \cdots & a_{2n-2} & u^{n-1}
\cr\noalign{\vskip 1mm}
a_n & a_{n+1} & \cdots & a_{2n-1} & u^n
}
\right|\,,
\qquad\qquad (n\ge 0),\\[3mm]
q_n(u)& =& \sum_{m=0}^n a_m {n \choose m} (-u)^{n-m},
\qquad\qquad (n \ge 0).\label{SMALL_Q}
\end{eqnarray}
\begin{theorem}\label{TH1}
The following identity holds for all integer values 
of $l$ and $n$, $l\ge 1, n\ge 1$: 
\[
\left|
\matrix{
p_l(u) & p_{l+1}(u)&\cdots & p_{l+n-1}(u)
\cr\noalign{\vskip 1mm}
p'_l(u) & p'_{l+1}(u)&\cdots & p'_{l+n-1}(u)
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots
\cr\noalign{\vskip 1mm}
p^{(n-1)}_l(u) & p^{(n-1)}_{l+1}(u)&\cdots & p^{(n-1)}_{l+n-1}(u)
} \right|
=
C_{l,n}
\left|
\matrix{ q_n(u) & q_{n+1}(u)&\cdots & q_{n+l-1}(u)
\cr\noalign{\vskip 1mm}
q_{n+1}(u) & q_{n+2}(u) &\cdots & q_{n+l}(u)
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots
\cr\noalign{\vskip 1mm}
q_{n+l-1}(u) & q_{n+l}(u)&\cdots & q_{n+2l-2}(u)
} \right|\,,
\]
where $C_{l,n}$ is independent of $u$:
\[
C_{l,n} = (-1)^{nl}\,
\prod_{k=1}^{n-1} k! \,
\left|
\matrix{ a_0 & a_1&\cdots & a_{k+l-1} 
\cr\noalign{\vskip 1mm}
a_1 & a_2 &\cdots & a_{k+l} 
\cr\noalign{\vskip 1mm} 
\vdots & \vdots & & \vdots 
\cr\noalign{\vskip 1mm} 
a_{k+l-1} & a_{k+l}&\cdots &  a_{2k+2l-2}
} \right|\,. 
\]
\end{theorem}
%
Now, as is well known (see \cite{Sze2}, p.27), if
\begin{equation}\label{MOM}
a_k = \int_a^b u^k w(u)\, du
\end{equation}
is the $k$th moment of the weight function $w$ on $]a,b[$,
then $p_n$ is the $n$th orthogonal polynomial 
associated with $w$ (up to normalization).
Thus Theorem~\ref{TH1} states that the Wronskian of $n$ consecutive 
orthogonal polynomials $p_l,\ldots ,p_{l+n-1}$
is proportional to a $l\times l$ Hankel determinant
of polynomials $q_n,\ldots , q_{n+2l-2}$ defined
in a simple way from the moments of the $p_k$'s.
In fact, it is easy to see 
%using Lemma~\ref{LEMPHI} below
that the $q_k$'s are given by the generating series
\begin{equation}\label{GENER}
\sum_{k\ge 0} q_k(u)\, {t^k\over k!}
=
e^{-tu}
\,
\int_a^b e^{tx} w(x) dx \,.
\end{equation}
%
The proof of Theorem~\ref{TH1} will be obtained 
by application of a `master identity' of Turnbull
on minors of a matrix \cite{Turn,Le1}.
%

Theorem~\ref{TH1} should be regarded as a limiting case of the
well-known result of Christoffel for the orthogonal polynomials
associated with the weight function
$(u-x_1)\cdots (u-x_n)w(u)$
(see \cite{Sze2}, p. 30).
And indeed, one could also obtain this Theorem by taking an appropriate
limit in the formula
given by Lascoux and Shi He \cite{LSH} for the Christoffel polynomials 
(see below, end of Section~\ref{SECT-PROOF}).

Assuming Theorem~\ref{TH1}, the verification of the
identities of Karlin and Szeg\"o is therefore 
essentially reduced to the following algebraic property
of the classical polynomials, discovered by Burchnall \cite{Bu}.
%ten years before the appearence of \cite{KS}.
%
\begin{theorem} {\rm \cite{Bu}} \label{TH2}
Let $Q_n(x)$ denote one of the following classes of
polynomials:
\[
\mbox{\rm (i)  } Q_n(x) = P_n^{(\lambda)}(x)\,; \quad 
\mbox{\rm (ii)  } Q_n(x) = L_n^{(\alpha)}(x)\,; \quad 
\mbox{\rm (iii)  } Q_n(x) = H_n(x)\,.
\]
Then we have
\begin{equation}\label{Q_SMALL}
Q_n(x) = \lambda_n \, \phi(x)^n \, q_n(u)
\end{equation}
where $q_n(u)$ is of the form {\rm (\ref{SMALL_Q})},
and in case {\rm (i)}:
\[
\lambda_n = P_n^{(\lambda)}(1),\ \ 
\phi(x) = (x^2-1)^{1/2},\ \ 
u = {-x\over (x^2-1)^{1/2}},\ \ 
a_{2p} = {({1\over 2})_p\over (\lambda +{1\over 2})_p}, \ \
a_{2p+1} = 0,
\]
in case {\rm (ii)}:
\[
\lambda_n =  L_n^{(\alpha)}(0),\quad   
\phi(x) = -x,\quad 
u = 1/x,\quad     
a_m = {1\over (\alpha + 1)_m},
\]
and in case {\rm (iii)}:
\[ 
\lambda_n =  (-2)^n,\quad   
\phi(x) = 1,\quad   
u = x,\quad      
a_{2p} = {(-1)^p\,(2p)!\over 2^{2p}\,p!}, \quad
a_{2p+1}=0.
\]
\end{theorem}
Theorem~\ref{TH2} is rather straightforward to check, using the differential
relation satisfied by each class of polynomials.
On the other hand, the differential relation for
general Jacobi polynomials does not allow to write them in the
form (\ref{Q_SMALL}), which explains why there is 
no identity of the Karlin-Szeg\"o type for these polynomials.
One may note that Burchnall had already observed that 
Tur\'an's inequality~(\ref{TURAN}) follows directly from Theorem~\ref{TH2},
as well as its generalization to polynomials of the classes 
(i), (ii), (iii).
He had also given formula~(12.1) of \cite{KS} for $n=0,1$.

Using the results of Karlin and Szeg\"o
on Wronskians of orthogonal polynomials (\cite{KS}, p.6)
one deduces immediately from Theorem~\ref{TH1} the following
%
\begin{corollary}\label{COR}
Let $w$ be an arbitrary non-negative weight function on a real interval
$]a,b[$, and let $q_n(u)$ be defined by {\rm (\ref{GENER})}. 
For $l,n \ge 1$, set 
\[
T(w;l,n;u) = 
\left|
\matrix{ q_n(u) & q_{n+1}(u)&\cdots & q_{n+l-1}(u)
\cr\noalign{\vskip 1mm}
q_{n+1}(u) & q_{n+2}(u) &\cdots & q_{n+l}(u)
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots
\cr\noalign{\vskip 1mm}
q_{n+l-1}(u) & q_{n+l}(u)&\cdots & q_{n+2l-2}(u)
} \right|\,.
\]
Then, if $n$ is even, $T(w;l,n;u)$ keeps a constant sign for
all real $u$, and if $n$ is odd, $T(w;l,n;u)$ has exactly
$l$ real simple zeros strictly interlaced between the $l+1$ zeros of 
$T(w;l+1,n;u)$.
\end{corollary}
%
By Theorem~\ref{TH1}, when $n=1$ the polynomial $T(w;l,1;u)$ is 
equal up to a numerical factor to the $l$th orthogonal polynomial $p_l(u)$
associated with $w$. 
Thus Corollary~\ref{COR} is a generalization of the well-known
fact that $p_l(u)$ has $l$ real simple zeros interlaced between
the zeros of $p_{l+1}(u)$.

The paper is organized as follows.
Section~\ref{SECT-DISF} provides some background on
determinantal identities. 
It also introduces the Schur function notation for
orthogonal polynomials, which is extremely convenient
for handling formulas such as Theorem~\ref{TH1}.
The proof of Theorem~\ref{TH1} is given in Section~\ref{SECT-PROOF},
and its specialization to the Karlin-Szeg\"o identities is considered
in Section~\ref{SECT-SPEC}. 
To make the paper self-contained, we have included a proof 
of Theorem~\ref{TH2}. 
Section~\ref{SECT-MIS} gives examples of specializations of 
Theorem~\ref{TH1} to other classes of polynomials,
like Euler polynomials or Bernoulli polynomials.
Finally, Section~\ref{SECT-OTHER} discusses
another family of identities of Karlin and Szeg\"o also contained in \cite{KS}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  SECTION 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Determinantal identities and Schur functions}
\label{SECT-DISF}

We begin by reviewing briefly the notation of \cite{Le1}
for minor identities, which is a variant of Turnbull's dot
notation (see \cite{Turn}, p.27).
Let $M$ be a $n \times p$ matrix with $p > n$, 
and $a,b,\ldots,c$ be $n$ column vectors of $M$. 
The maximal minor of $M$ taken on these $n$ columns 
is denoted by either a bracket or a one line tableau:
\[
[ab\ldots c] = \tab{a &b &\ldots &c\cr}\,.
\]
A product of $k$ minors of $M$ is designated by a 
$k \times n$ tableau. 
Thus for $k=3$,
$$[ab\ldots c].[de\ldots f].[gh\ldots i] =
\tab{
a &b &\ldots &c\cr\th
d &e &\ldots &f\cr\th
g &h &\ldots &i\cr
}\,.$$
To denote alternating sums of products of minors, 
we use tableaux with boxes enclosing certain vectors. 
Let $T$ be a $k \times n$ tableau and $A$ a subset of elements of $T$. 
Write $i_j$ for the number of elements of $A$ lying
in the $j$th row of $T$.
The tableau $\tau$ obtained from $T$ by boxing
all elements of $A$ will be used to denote the sum
\begin{equation}\label{BOXES}
\tau = {1\over i_1!\cdots i_k!} \sum_{\sigma \in \SG(A)}
\sgn (\sigma) \, \sigma(T)\,,
\end{equation}
where $\sigma(T)$ is the tableau in
which the elements of $A$ are permuted by $\sigma$.
Due to the skew-symmetry of 
$(a,b,\ldots , c) \mapsto [ab\cdots c]$, if $\sigma$
permutes between themselves the elements of $A$
lying in the $j$th row for all $j$, then clearly
$\sgn (\sigma) \, \sigma(T) = T$.
Such permutations $\sigma$ form a Young subgroup $\SG_T$ of $\SG(A)$ 
of cardinality $ i_1!\cdots i_k!$. 
Hence (\ref{BOXES}) may be rewritten as  
\begin{equation}
\tau = \sum_{\sigma }
\sgn (\sigma) \, \sigma(T)\,,
\end{equation}
where $\sigma$ ranges now over a set of representatives of
the left cosets of $\SG_T$ in $\SG(A)$.
For example,
\[
\tau\ =\ \tab{
\bo{a}& \bo{b}& c\cr\th
d& e& \bo{f}\cr
}\ :=\
 \tab{
a& b& c\cr\th
d& e& f\cr
}\ -\
 \tab{
f& b& c\cr\th
d& e& a\cr
}\ -\
 \tab{
a& f& c\cr\th
d& e& b\cr
 } \,.
\]
We can now state Turnbull's identity 
(see \cite{Turn}, p. 48 and \cite{Le1}).
\begin{theorem}\label{TURN}
Let $\tau$ be a $p\times n$ tableau with set of enclosed elements $A$
of cardinality $\le n$.
Let $R$ be one of the rows of $T$ and denote by $B$ the set of
elements of $R$ which are not enclosed. 
Form a new tableau $\upsilon$ by {\rm (i)} exchanging the elements 
of $A$ which do not belong to $R$ with elements of $B$;
{\rm (ii)} removing the boxes of the elements of $A$; 
{\rm (iii)} boxing the elements of $A$; then, \[\tau = \upsilon\, .\]
\end{theorem}
Taking for instance 
\[
\tau=\tab{
\bo{\bf a}& \alpha& \beta& \gamma& \delta& \varepsilon\cr\th
\bo{\bf b}& \bo{\bf c}& f& g&h& i\cr\th
\bo{\bf d}& j& k& l& m& o\cr
} 
\]
and choosing for $R$ the first row, we have 
$A=\{ \bf a\rm,\bf b\rm,\bf c\rm,\bf d \rm\}$,
$B=\{ \alpha,\beta,\gamma,\delta,\varepsilon\}$,
and we obtain
\[
\tau=\tab{
\bo{\bf a}& \alpha& \beta& \gamma& \delta& \varepsilon\cr\th
\bo{\bf b}& \bo{\bf c}& f& g&h& i\cr\th
\bo{\bf d}& j& k& l& m& o\cr
} =
\tab{
{\bf a}& {\bf b}& {\bf c}& {\bf d}& \bo{\delta}& \bo{\varepsilon}\cr\th
\bo{\alpha}& \bo{\beta}& f& g& h& i\cr\th
\bo{\gamma}& j& k& l& m& o\cr
}=\upsilon\,.
\]
Note that here $\tau$ represents a sum of 12 products of minors,
whereas $\upsilon$ stands for a sum of 30 products. 
It has been shown in \cite{Le1} that many classical determinantal identities
are easily obtained as simple specializations of Theorem~\ref{TURN}.
Such identities would become quite cumbersome to state if 
one did not use a convenient system of notation showing clearly what
are the transformations being performed.

Similarly, as explained by Lascoux in \cite{La}, the Schur function 
approach to orthogonal polynomials  simplifies
greatly the algebraic aspects of this subject.
Recall that the complete homogeneous symmetric functions
$S_i(E)$ of a set $E$ of indeterminates are defined
via the generating series
\begin{equation}\label{INDETER}
\sigma(E,t):=\sum_{i\ge 0} S_i(E) \, t^i
=
\prod_{e\in E} {1\over 1-te}\,.
\end{equation}
In particular $S_0(E) = 1$.
We decide that for $i<0$, $S_i(E) = 0$.
If $F$ is a second set of indeterminates, we define the formal
sum and difference $E+F$ and $E-F$ by
\begin{eqnarray}
\sigma(E+F,t) &:= & \sigma(E,t)\,\sigma(F,t)\,, \label{SUM}\\
\sigma(E-F,t) &:= & {\sigma(E,t)\over \sigma(F,t)} \,.\label{DIF}
\end{eqnarray}
As an example, consider the case when $E$ consists of only
one variable $E=\{x\}$, and $F$ of $n$ variables.
We have
\[
\sigma(x-F,t) = {\prod_{f\in F} (1-tf) \over 1-tx}
= \prod_{f\in F} (1-tf) \,\sum_{k\ge 0} t^kx^k
\]
so that 
\[
S_n(x-F) =  \prod_{f\in F} (x-f)
\]
is the monic polynomial with set of roots $F$.

An important idea, extensively used by Littlewood (see \cite{Li}, chap. 7),
is that any sequence $a_k, \ k\ge 1$ of elements of a commutative ring $R$
can be regarded as the sequence of complete symmetric functions
of a fictitious set of variables $E$:
\[
S_k(E) = a_k\,.
\]
Indeed it is well known that the $S_k$ form a set of algebraically
independent generators of the ring of symmetric functions
(see \cite{Mcd}), and thus one can always define a homomorphism
from this ring to $R$ by assigning
$S_k \mapsto S_k(E) := a_k$.
Of course this is very formal, but it allows to understand
that certain identities, between orthogonal polynomials
for instance, arise naturally as specializations of
identities at the level of symmetric functions. 
An example of this phenomenon was given in \cite{LT}
where a conjecture of Favreau for the computation of the
linearization coefficients of Bessel polynomials 
was shown to result from a known formula for multiplying
two staircase Schur functions.

In the sequel the notation $S_k(E)$ will therefore
indicate nothing but a certain specialization of the
ring of symmetric functions. 
In this context, it is customary to call $E$ an alphabet.
A set of indeterminates is regarded as a particular 
case of alphabet by means of (\ref{INDETER}).
The symmetric functions of a sum $E+F$ or a difference
$E-F$ of alphabets are defined via (\ref{SUM}) and (\ref{DIF}).

Given $n$ alphabets $E_1, \ldots ,E_n$ and a sequence 
$I=(i_1,\ldots ,i_n)$ of integers, define the 
(multi) Schur function
\begin{equation}
S_I(E_1,\ldots ,E_n) = \det[S_{i_l+l-k}(E_l)]_{1\le k,l \le n}\,.
\end{equation}
We shall often use the exponential notation for sequences
$I$ with repeated parts, and write 
$I=(i_1^{m_1} i_2^{m_2} \cdots i_r^{m_r})$
to indicate the sequence with $m_1$ terms equal to $i_1$,
$m_2$ terms equal to $i_2$ and so on.

As recalled in Section~\ref{SECT1}, orthogonal polynomials
have a determinantal expression which can be seen as a particular
instance of Schur function.
Indeed, putting
\[
S_k(E) = {a_k\over a_0}\,,\qquad 
a_k = \int_a^b u^k w(u)\, du\,,
\]
we see that 
\begin{equation}
S_{(n,\ldots,n,0)}(E,\ldots ,E,u)
=
\left|
\matrix{
S_n(E) & S_{n+1}(E) & \cdots & S_{2n-1}(E) & u^n 
\cr\noalign{\vskip 1mm}
S_{n-1}(E) & S_n(E) & \cdots & S_{2n-2}(E) & u^{n-1}
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots &\vdots 
\cr\noalign{\vskip 1mm}
S_1(E) & S_2(E) & \cdots & S_n(E) & u
\cr\noalign{\vskip 1mm}
S_0(E) & S_1(E) & \cdots & S_{n-1}(E) & 1
}
\right|
\end{equation}
is up to a scalar the $n$th orthogonal polynomial
associated with the weight function $w$.
Here, we consider the single variable $u$ as a particular alphabet 
by defining
\[
\sigma(u,t)=\sum_{i\ge 0} S_i(u) \, t^i
:=
{1\over 1-tu}
=
\sum_{i\ge 0} u^i t^i\,.
\]
By subtraction of rows in this determinant we arrive at the following
more symmetric expression:
\begin{eqnarray*}
S_{(n^n\,0)}(E,\ldots ,E,u)
&=&
\left|
\matrix{
S_n(E-u) & S_{n+1}(E-u) & \cdots & S_{2n-1}(E-u) 
\cr\noalign{\vskip 1mm}
S_{n-1}(E-u) & S_n(E-u) & \cdots & S_{2n-2}(E-u)
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots 
\cr\noalign{\vskip 1mm}
S_1(E-u) & S_2(E-u) & \cdots & S_n(E-u) 
}
\right|\\[2mm]
&=&
S_{(n^n)}(E-u,\ldots ,E-u)\,.
\end{eqnarray*}
This is now an ordinary Schur function (\ie depending
on a single alphabet $E-u$) that we denote more concisely
by $S_{(n^n)}(E-u)$. 

The formula $S_{(n^n\,0)}(E,\ldots ,E,u)
=S_{(n^n)}(E-u)$ is in fact a special case of a very
useful lemma going back to Jacobi (see \cite{Ja}, p.371).
%
\begin{lemma}\label{JACOBI}
Let $F$ be an alphabet such that $S_k(-F)=0$ for $k>m$.
Then 
\[
S_I(E_1,\ldots ,E_n)=
\left|\matrix{
S_{i_1}(E_1-F)&\ldots &S_{i_n+n-1}(E_n-F)\cr
\vdots& &\vdots\cr
S_{i_1-n+m+1}(E_1-F)&\ldots &S_{i_n+m}(E_n-F)\cr
S_{i_1-n+m}(E_1)&\ldots &S_{i_n+m-1}(E_n)\cr
\vdots & &\vdots\cr
S_{i_1-n+1}(E_1)&\ldots &S_{i_n}(E_n)\cr
}\right|\,.
\]
\end{lemma}
%
Indeed, we have $S_k(E_j-F)=
S_k(E_j)+S_{k-1}(E_j)S_1(-F)+\ldots + S_1(E_j)S_{k-1}(-F) + S_k(-F).$
Thus, since $S_k(-F)=0$ for $k>m$, the determinant of the right-hand
side is obtained from that of the left-hand side by
adding to each of the first $n-m$ rows a linear combination of the next $m$ rows.

Note that if $F$ is a set of $m$ indeterminates, then it satisfies
the condition $S_k(-F)=0$ for $k>m$, since by definition
\[
\sum_{i\ge 0} S_i(-F) \, t^i
=
\prod_{f\in F} (1-tf)\,.
\]

Finally, as a further example of this alphabet notation,
let us consider again the alphabet $u$ consisting
of the single variable $u$.
Iterating (\ref{SUM}), we define a new alphabet $nu:=u+\cdots +u$
by
\[
\sum_{i\ge 0} S_i(nu) \, t^i
:=
\sigma(u,t)^n
=
{1\over (1-tu)^n}
\]
so that 
$S_k(nu) = {n+k-1\choose k} u^k$.
Similarly, we set
\[ 
\sum_{i\ge 0} S_i(-nu) \, t^i
:= 
\sigma(u,t)^{-n}
= (1-tu)^n\,,
\]
which gives 
$S_k(-nu) = {n\choose k} (-u)^k$.
Now one can write for an arbitrary alphabet $E$:
\begin{equation}\label{PHI}
S_n(E-nu) = \sum_{k=0}^n S_k(E)\,{n\choose k} (-u)^{n-k},
\end{equation}
in which we recognize the polynomial $q_n(u)$ of (\ref{SMALL_Q})
with $S_k(E) = a_k/a_0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  SECTION 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of Theorem~1} 
\label{SECT-PROOF}
Let $E$ be an arbitrary alphabet.
We put
\begin{eqnarray*}
p_m(u) & = & (-1)^{{m+1\choose 2}}\,S_{(m^m)}(E-u) = 
 (-1)^{{m+1\choose 2}}\,S_{(m^m0)}(E,\ldots ,E,u) \,,\\[2mm]
q_m(u) & = &S_m(E-mu)\,.
\end{eqnarray*}
We first remark that 
\begin{equation}\label{REDUCTION}
\left|
\matrix{ q_{n+l-1}(u) & q_{n+l}(u)&\cdots & q_{n+2l-2}(u)
\cr\noalign{\vskip 1mm}
q_{n+l-2}(u) & q_{n+l-1}(u) &\cdots & q_{n+2l-3}(u)
\cr\noalign{\vskip 1mm}
\vdots & \vdots & & \vdots
\cr\noalign{\vskip 1mm}
q_n(u) & q_{n+1}(u)&\cdots & q_{n+l-1}(u)
} \right|
=
S_{((n+l-1)^l)}(E-nu) \,.
\end{equation}
Indeed, since all the terms of the sequence
$I=((n+l-1)^l)$ are equal,
we can use repeatedly Lemma~\ref{JACOBI} both in the rows and
in the columns of the Schur function $S_{((n+l-1)^l)}(E-nu)$
and subtract the alphabet $ku$ in the $(k+1)$th column
and the $(l-k)$th row to get the determinant in the
left-hand side of (\ref{REDUCTION}).

Denoting by $\Wr(f_1(u),\ldots ,f_k(u))$ the Wronskian of
the functions $f_j(u),\ 1\le j \le k$, we deduce from Section~\ref{SECT-DISF}
that the formula to be proved can be rewritten as
\begin{equation}\label{TO_PROVE}
\Wr(S_{(l^l)}(E-u),\ldots ,S_{((l+n-1)^{l+n-1})}(E-u))
=
C_{l,n}(E) \, S_{((l+n-1)^l)}(E-nu) \,,
\end{equation}
where 
\begin{equation}\label{CONSTANT}
C_{l,n}(E) = (-1)^{{n\choose 2}}
\prod_{k=1}^{n-1} k!\,
S_{((l+k-1)^{l+k})}(E) \,.
\end{equation}

Let us introduce the $(l+n)\times (l+3n-2)$ matrix
\[
\left[
\matrix{
1 & 0 & \ldots & 0 & S_{l+n-1} & S_{l+n} & \ldots & S_{2l+2n-3} &
u^{l+n-1} & {du^{l+n-1}\over du} & \ldots & {d^{n-1}u^{l+n-1}\over du^{n-1}} 
\cr\noalign{\vskip 1mm}
%
0 & 1 & \ldots & 0 & S_{l+n-2} & S_{l+n-1} & \ldots & S_{2l+2n-4} &
u^{l+n-2} & {du^{l+n-2}\over du} & \ldots & {d^{n-1}u^{l+n-2}\over du^{n-1}} 
\cr\noalign{\vskip 1mm}
%
\vdots & \vdots &\ddots & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots
\cr\noalign{\vskip 1mm}
%
0 & 0 & \ldots & 1 & S_{l+1} & S_{l+2} & \ldots & S_{2l+n-1} &
u^{l+1} & {du^{l+1}\over du} & \ldots & {d^{n-1}u^{l+1}\over du^{n-1}} 
\cr\noalign{\vskip 1mm}
%
0 & 0 & \ldots & 0 & S_l & S_{l+1} & \ldots & S_{2l+n-2} &
u^l & {du^l\over du} & \ldots & {d^{n-1}u^l\over du^{n-1}} 
\cr\noalign{\vskip 1mm}
%
\vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots
\cr\noalign{\vskip 1mm}
%
0 & 0 & \ldots & 0 & S_1 & S_2 & \ldots & S_{l+n-1} &
u & 1 & \ldots & 0 
\cr\noalign{\vskip 1mm}
%
0 & 0 & \ldots & 0 & S_0 & S_2 & \ldots & S_{l+n-2} &
1 & 0 & \ldots & 0 
}\right]
\]
where $S_k$ is short for $S_k(E)$.
The column vectors of this matrix will be denoted from left to right by:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  local macros
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\1{{\bf 1}}
\def\2{{\bf 2}}
\def\nmu{{\bf n-1}}
\def\nmd{{\bf n-2}}
\def\sz{\sigma_0}
\def\su{\sigma_1}
\def\sd{\sigma_2}
\def\sl{\sigma_l}
\def\slpu{\sigma_{l+1}}
\def\slmu{\sigma_{l-1}}
\def\slnmu{\sigma_{l+n-1}}
\def\slnmd{\sigma_{l+n-2}}
\def\u{{\bf u}}
\def\up{{\bf u'}}
\def\unmu{{\bf u^{(n-1)}}}
\def\unmd{{\bf u^{(n-2)}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[
\1 , \2 , \ldots , \nmu , \sz , \su , \ldots ,
\slnmd , \u , \up , \ldots , \unmu\,.
\]
Using the notation of Section~\ref{SECT-DISF}, the left-hand side of 
(\ref{TO_PROVE}) is written as
\[
\left|
\matrix{
[\1\2\ldots \nmu \sz \su \ldots \slmu\u]
&
[\1\2\ldots \nmd \sz \su \ldots \sl\u]
&
\ldots
&
[\sz \ldots \slnmd\u]
\cr\noalign{\vskip 1mm}
[\1\2\ldots \nmu \sz \su \ldots \slmu\up]
&
[\1\2\ldots \nmd \sz \su \ldots \sl\up]
&
\ldots
&
[\sz \ldots \slnmd\up]
\cr\noalign{\vskip 1mm}
\vdots &\vdots & & \vdots
\cr\noalign{\vskip 1mm}
[\1\ldots \nmu \sz  \ldots \slmu\unmu]
&
[\1\ldots \nmd \sz  \ldots \sl\unmu]
&
\ldots
&
[\sz \ldots \slnmd\unmu]
}\right|
\]
\[
\qquad\qquad\qquad\qquad = \quad
\tab{
\1 & \2 & \ldots & \nmd & \nmu & \sz & \su & \ldots & \slmu & \bo{\u} \cr\th
\1 & \2 & \ldots & \nmd & \sz & \su &\sd & \ldots & \sl & \bo{\up} \cr\th
\vdots &\vdots & &     &      &     &     &        &     & \vdots   \cr\th
\1 & \sz & \ldots& \ldots & \ldots  &\ldots & \ldots & \ldots & \slnmu & \bo{\unmd}\cr\th
\sz & \su & \ldots&\ldots&\ldots &\ldots &\ldots  & \ldots & \slnmd & \bo{\unmu}
}
\]
Now using the transformation of Theorem~\ref{TURN} with
$R$ being the last row, this tableau is equal to
\[
\tab{
\1 & \2 & \ldots & \nmu & \nmu & \sz & \su & \ldots & \slmu & \bo{\sl} \cr\th
\1 & \2 & \ldots & \nmd & \sz & \su & \sd & \ldots & \sl & \bo{\slpu} \cr\th
\vdots &\vdots &      &      &     &     &   &     &     & \vdots   \cr\th
\1 & \sz & \ldots& \ldots& \ldots    &\ldots &\ldots & \ldots & \slnmu & \bo{\slnmd}\cr\th
\bo{\sz} & \bo{\su} &\ldots & \ldots &\bo{\slmu} &\u &\up    & \ldots & \unmd & \unmu
}\qquad\qquad\qquad\qquad
\]
\medskip
\[
\qquad\qquad\qquad\qquad
= \quad
\tab{
\1 & \2 & \ldots & \nmu & \nmu & \sz & \su & \ldots & \slmu & \sl \cr\th
\1 & \2 & \ldots & \nmd & \sz & \su & \sd & \ldots & \sl & \slpu \cr\th
\vdots &\vdots &      &      &     &     &   &     &     & \vdots   \cr\th
\1 & \sz & \ldots& \ldots& \ldots    &\ldots &\ldots & \ldots & \slnmu & \slnmd\cr\th
\sz & \su &\ldots & \ldots &\slmu &\u &\up    & \ldots & \unmd & \unmu
}
\]
Note that all boxes have been deleted in the second tableau because
all the other permutations of the letters enclosed give rise to tableaux
with two equal letters on some row, which are therefore equal to
zero in view of the skew-symmetry of the determinant.
Thus, we have rewritten the Wronskian of (\ref{TO_PROVE}) as 
a product of $n$ determinants, where only the last one depends on the
variable $u$.
Explicitly, we have obtained that the left-hand side of (\ref{TO_PROVE})
is equal to
\[
S_{(l^{l+1})}(E)\,S_{((l+1)^{l+2})}(E)\,\cdots
S_{((l+n-2)^{l+n-1})}(E)\,\Delta(u) \,,
\]
where
\[
\Delta(u)
=
\left|
\matrix{
S_{l+n-1}(E) & S_{l+n}(E) & \ldots & S_{2l+n-2}(E) &
u^{l+n-1} & {du^{l+n-1}\over du} & \ldots & {d^{n-1}u^{l+n-1}\over du^{n-1}}
\cr\noalign{\vskip 1mm}
%
S_{l+n-2}(E) & S_{l+n-1}(E) & \ldots & S_{2l+n-3}(E) &
u^{l+n-2} & {du^{l+n-2}\over du} & \ldots & {d^{n-1}u^{l+n-2}\over du^{n-1}}
\cr\noalign{\vskip 1mm}
%
\vdots & \vdots & & \vdots & \vdots &  \vdots  & & \vdots
\cr\noalign{\vskip 1mm}
%
S_1(E) & S_2(E) & \ldots & S_l(E) &
u & 1 & \ldots & 0
\cr\noalign{\vskip 1mm}
%
S_0(E) & S_1(E) & \ldots & S_{l-1}(E) &
1 & 0 & \ldots & 0 
}\right|\,.
\]
Now, noting that 
\[
{d^ku^m\over du^k} = k!\,{m\choose k} u^{m-k} = k!\,S_{m-k}\left((k+1)u\right)\,,
\]
we have 
\[
\Delta(u) =
1!\,2!\,\cdots \, (n-1)! \,
\left|
\matrix{
S_{l+n-1}(E) &  \ldots & S_{2l+n-2}(E) &
S_{l+n-1}(u) & \ldots & S_l(nu)  
\cr\noalign{\vskip 1mm}
%
\vdots & & \vdots & \vdots & & \vdots 
\cr\noalign{\vskip 1mm}
%
S_0(E) &  \ldots & S_{l-1}(E) &
S_0(u) & \ldots & S_{-n+1}(nu)
}\right|\,.
\]
Finally, using Lemma~\ref{JACOBI} to subtract the alphabet $nu$ in the
$l$ first rows of this last determinant, we see that it reduces to
\[
\left|
\matrix{
S_{l+n-1}(E-nu) &  \ldots & S_{2l+n-2}(E-nu) &
0 & \ldots & 0 & 0 
\cr\noalign{\vskip 1mm}
%
\vdots & \ddots & \vdots & \vdots & & \vdots & \vdots
\cr\noalign{\vskip 1mm}
%
S_n(E-nu) &  \ldots & S_{l+n-1}(E-nu) &
0 & \ldots & 0 & 0 
\cr\noalign{\vskip 1mm}
%
S_{n-1}(E) &  \ldots & S_{l+n-2}(E) &
 S_{n-1}(u) & \ldots &S_1((n-1)u)  & 1 
\cr\noalign{\vskip 1mm}
%
S_{n-2}(E) &  \ldots & S_{l+n-3}(E) &
S_{n-2}(u) & \ldots & 1 & 0 
\cr\noalign{\vskip 1mm}
%
\vdots & & \vdots & \vdots &\adots & \vdots & \vdots
\cr\noalign{\vskip 1mm}
%
S_0(E) &  \ldots & S_{l-1}(E) &
1  & \ldots & 0 & 0
}\right|
\]
\[
=\quad (-1)^{{n\choose 2}}\ 
\left|
\matrix{
S_{l+n-1}(E-nu) &  \ldots & S_{2l+n-2}(E-nu) 
\cr\noalign{\vskip 1mm}
%
\vdots & & \vdots 
\cr\noalign{\vskip 1mm}
%
S_n(E-nu) &  \ldots & S_{l+n-1}(E-nu) 
}\right|\,
\]
so that 
\[
\Delta(u)=(-1)^{{n\choose 2}}\,1!\,2!\,\cdots \, (n-1)!
\, S_{((l+n-1)^l)}(E-nu)\,,
\]
as required.
This finishes the proof of Theorem~\ref{TH1}.

\bigskip
We end this section by noting the closely related expression 
given by Lascoux and Shi He \cite{LSH}
of the Christoffel polynomials
\[
\psi_{l,n} := \det[S_{((l+i-1)^{l+i-1})}(E-f_j)]_{1\le i,j \le n}\,,
\]
where $F=\{f_1,\ldots ,f_n\}$ is a set of $n$ indeterminates
(see \cite{Sze2}, p.30).
Lascoux and Shi He found that
\begin{equation}\label{GDC}
\psi_{l,n}
=
\prod_{1\le i,j \le n} (f_i-f_j)\,
\prod_{1\le j \le n-1}
S_{((l+j-1)^{l+j})}(E)
\ S_{((l+n-1)^l)}(E-F)\,.
\end{equation}
Theorem~\ref{TH1} may be regarded as the
limiting case $f_i \longrightarrow u$ of (\ref{GDC}). 
Indeed, $\psi_{l,n}$ is clearly
skew-symmetric in the $f_i$, and therefore is divisible
by the Vandermonde determinant
$\prod_{1\le i,j \le n} (f_i-f_j)$. 
The quotient may be expressed 
as a discrete Wronskian, that is, a Wronskian determinant
where derivatives are replaced by divided differences.
Once this division is performed, one can actually
pass to the limit $f_i \longrightarrow u$ 
and recover the usual Wronskian of Theorem~\ref{TH1}.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  SECTION 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Specialization to classical orthogonal polynomials}
\label{SECT-SPEC}
In the derivation of the Karlin-Szeg\"o
identities, the following lemma, 
which follows immediately from (\ref{PHI}), will be used.
\begin{lemma}\label{LEMPHI}
A sequence of functions $f_k(u)$ is of the form
\[
f_k(u) = S_k(E-ku) \,, \qquad (k\ge 0)
\]
for a certain alphabet $E$, if and only if $f_0\equiv 1$ and 
\[
{df_k(u)\over du} = -k f_{k-1}(u)\,, \qquad (k\ge 1)\,.
\]
In this case, the alphabet $E$ is specified by $S_k(E) = f_k(0)$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ultraspherical polynomials}

Let 
\begin{equation}\label{DEFU}
P_k^{(\lambda)}(x) =
\sum_{m=0}^{\lfloor {k\over 2}\rfloor}
(-1)^m \, {k-m+\lambda-1\choose k-m} {k-m\choose m} (2x)^{k-2m}
\end{equation}
denote the $k$th ultraspherical
polynomial with parameter $\lambda > -1/2$ (see \cite{Sze2}, (4.7.31)). 
(For $\lambda = 0$, the right-hand side of (\ref{DEFU}) is 0,
but $\lim_{\lambda\rightarrow 0}{{1\over \lambda}P_k^{(\lambda)}(x)}
= {2\over k}\,T_k(x)$ and the subsequent formulas remain valid
provided $P_k^{(\lambda)}(x)$ is replaced by this limit \cite{Sze2}, (4.7.8).)
We put
\[
f_k(u) = (x^2-1)^{-k/2}\, 
{P_k^{(\lambda)}(x)\over P_k^{(\lambda)}(1)}\,,
\]
where $u=-x(x^2-1)^{-1/2}$.
Then,
\begin{eqnarray*}
{df_k(u)\over du} &= &
{d\over dx}\left\{(x^2-1)^{-k/2} {P_k^{(\lambda)}(x)\over
 P_k^{(\lambda)}(1)}\right\}\,{dx\over du} \\[2mm]
& = &
(x^2-1)^{-(k-1)/2}
\left\{(x^2-1){d\over dx}
\left({P_k^{(\lambda)}(x)\over  P_k^{(\lambda)}(1)}\right) 
- kx {P_k^{(\lambda)}(x)\over  P_k^{(\lambda)}(1)}\right\}\,.
\end{eqnarray*}
Now, using the classical differential relation (\cite{Sze2}, (4.7.27))
\[
(1-x^2){d\over dx} \left({P_k^{(\lambda)}(x)\over P_k^{(\lambda)}(1)}\right)
= -kx{P_k^{(\lambda)}(x)\over P_k^{(\lambda)}(1)}
 + k{P_{k-1}^{(\lambda)}(x)\over P_{k-1}^{(\lambda)}(1)} \,,
\]
we obtain
\[
{df_k(u)\over du} = -k (x^2-1)^{-(k-1)/2} 
{P_{k-1}^{(\lambda)}(x)\over P_{k-1}^{(\lambda)}(1)}
= -k \,f_{k-1}(u) \,.
\]
Thus, by Lemma~\ref{LEMPHI}, $f_k(u)=S_k(\UU_\lambda -ku)$ 
for the alphabet $\UU_\lambda$ specified by
\begin{equation}\label{UUDEF}
S_k(\UU_\lambda) = f_k(0) = (-1)^{-k/2}
{P_k^{(\lambda)}(0)\over P_k^{(\lambda)}(1)}
=
\left\{
\matrix{\displaystyle{({1\over 2})_l\over (\lambda +{1\over 2})_l }
&\mbox{if $k=2l$,}\cr\noalign{\vskip 2mm}
0 & \mbox{if $k=2l+1$,}
}\right.
\end{equation}
where we have used Pochammer's symbol
$(a)_n:=a(a+1)\cdots (a+n-1)$.
Thus, Theorem~\ref{TH2} is verified in case (i).
%
It follows that 
\begin{eqnarray*}
\det\left[
{P_{n+i+j}^{(\lambda)}(x)\over P_{n+i+j}^{(\lambda)}(1)}
\right]_{0\le i,j \le l-1}
&=&
\det\left[
(x^2-1)^{(n+i+j)/2}\,S_{n+i+j}(\UU_\lambda - (n+i+j)u)
\right]_{0\le i,j \le l-1}  \\[2mm]
&=&
(x^2-1)^{l(n+l-1)/2}
\det\left[
S_{n+i+j}(\UU_\lambda - (n+i+j)u)
\right]_{0\le i,j \le l-1}  \\[2mm]
&=&
(-1)^{{l\choose 2}}
(x^2-1)^{l(n+l-1)/2}
S_{((n+l-1)^l)}(\UU_\lambda - nu) \,.
\end{eqnarray*}

On the other hand, the moments of the weight function
\[
w_\lambda(u) = (1-u^2)^{\lambda - 1/2}\,, \qquad (-1<u<1)
\]
associated with the polynomials $P_n^{(\lambda)}(u)$ are readily computed, 
and one finds
\[
{\int_{-1}^1 u^k\,w_\lambda(u)du \over
\int_{-1}^1 w_\lambda(u) du}
=
\left\{
\matrix{\displaystyle{({1\over 2})_l\over (\lambda +1)_l }
&\mbox{if $k=2l$,}\cr\noalign{\vskip 2mm}
0 & \mbox{if $k=2l+1$.}
}\right.
\]
Thus, comparing with (\ref{UUDEF}), we have
\begin{equation}\label{MOMULTRA}
{\int_{-1}^1 u^k\,w_\lambda(u)du \over
\int_{-1}^1 w_\lambda(u) du}
 = S_k(\UU_{\lambda + 1/2})\,.
\end{equation}
Note the shift $\lambda \longrightarrow \lambda +1/2$ which explains 
in particular the relationship between Legendre and Tchebichev
polynomials expressed by (\ref{KSZ1}).
It follows from (\ref{MOMULTRA}) that 
the Schur function $S_{(n^n)}(\UU_\lambda-u)$
is equal up to a numerical factor to 
$P_n^{(\lambda-1/2)}(u)$.
Therefore, applying Theorem~\ref{TH1} we find that
\[
\det\left[
{P_{n+i+j}^{(\lambda)}(x)\over P_{n+i+j}^{(\lambda)}(1)}
\right]_{0\le i,j \le l-1}
= \ 
A_{l,n}^{(\lambda)} \, (x^2-1)^{l(n+l-1)/2} \,
\Wr(P_{l}^{(\lambda-1/2)}(u),\ldots , P_{l+n-1}^{(\lambda-1/2)}(u))\,,
\]
where $A_{l,n}^{(\lambda)}$ is a constant depending only on $l,n$ and $\lambda$.
By (\ref{CONSTANT}), to evaluate this constant it remains to compute the 
specialized Schur functions $S_{(m^{m+1})}(\UU_\lambda)$.
We omit this calculation and only mention that it
may be done using the following result of Saalsch\"utz
\cite{Saa}:
\begin{lemma}\label{SAAL}
Write $(2m+1)!!=(2m+1)(2m-1)\ldots 3.1$. Then
\[
\left|
\matrix{
(2k+1)!! & (a)_1\,(2k-1)!! & \ldots & (a)_{n-1}\,(2(k-n)+3)!! 
\cr\noalign{\vskip 1mm}
%
(2k+3)!! & (a+1)_1\,(2k+1)!!& \ldots & 
(a+1)_{n-1}\,(2(k-n)+5)!!
\cr\noalign{\vskip 1mm} 
\vdots & \vdots  &  & \vdots 
\cr\noalign{\vskip 1mm}
(2(k+n)-1)!! & (a+n-1)_1\,(2(k+n)-3)!!  &\ldots & 
(a+n-1)_{n-1}\,(2k+3)!!
}\right|
\qquad
\]
\[
=
(2(k-a)+1)^{n-1}(2(k-a)-1)^{n-2}\cdots (2(k-a-n)+5)
(2k+1)!!\cdots (2(k-n)+3)!!
\]
\end{lemma} 
This completes our derivation of the identity of Karlin and Szeg\"o 
for ultraspherical polynomials (\cite{KS}, (14.1)).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Laguerre polynomials}

The (generalized) Laguerre polynomials are given by (\cite{Sze2}, (5.1.6))
\begin{equation}
L_k^{(\alpha)}(x) = \sum_{m=0}^k (-1)^m
{k+\alpha \choose k-m} {1\over m!}\, x^m \,.
\end{equation}
Defining $\LL_\alpha$ by 
\begin{equation}
S_m(\LL_\alpha) = {1\over (1+\alpha)_m}\,, \qquad (m\ge 0)
\end{equation}
we obtain easily that
\begin{equation}
{L_k^{(\alpha)}(x)\over L_k^{(\alpha)}(0)}
= (-x)^k\,S_k(\LL_\alpha -ku),
\end{equation}
where $u=1/x$.

On the other hand, the moments of the weight function
\[
w_\alpha (u) = x^\alpha e^{-x}\,, \qquad (x>0)
\]
are given by
\[
{\int_{0}^{+\infty} u^k\,w_\alpha(u)du \over
\int_{0}^{+\infty} w_\alpha(u) du}
=
(1+\alpha)_k
\,.
\]
It follows by a simple calculation that the polynomial
$S_{(m^m)}(\LL_\alpha - u)$ is equal up to a numerical factor
to $x^{-m}\,L_m^{(-\alpha - 2m)}(-x)$, and thus, by application
of Theorem~\ref{TH1}, we obtain
\[
\det\left[
{L_{n+i+j}^{(\alpha)}(x)\over L_{n+i+j}^{(\alpha)}(0)}
\right]_{0\le i,j \le l-1}
= \
B_{l,n}^{(\alpha)} \, x^{l(n+l-1)} \,
\Wr(\lambda_l(u),\ldots , \lambda_{l+n-1}(u))\,,
\]
where
\[
\lambda_p(u) := u^p\,L_p^{(-\alpha-2p)}(-u^{-1}) \,.
\]
The value of $B_{l,n}^{(\alpha)}$ may be calculated from
(\ref{CONSTANT}) and the easily checked formula:
\[
S_{(m^{m+1})}(\LL_\alpha) =
\prod_{k=0}^m {(-1)^{m+1\choose 2}\,k!\over (1+\alpha)_{m+k}}
\,,
\]
and this completes our derivation of the identity of Karlin and Szeg\"o
for Laguerre polynomials (\cite{KS}, (16.1)).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hermite polynomials}
 
The Hermite polynomials are given by (\cite{Sze2}, (5.5.4))
\begin{equation}
H_k(x) = k!\,\sum_{m=0}^{\lfloor {k\over 2}\rfloor} (-1)^m
{1\over m!(k-2m)!}\, (2x)^{k-2m} \,.
\end{equation}
Setting
\begin{equation}\label{HHDEF}
S_m(\HH) = 
\left\{
\matrix{\displaystyle {(-1)^l\,(2l)!\over 2^{2l}\,l!}
&\mbox{if $m=2l$,}\cr\noalign{\vskip 2mm}
0 & \mbox{if $m=2l+1$,}
}\right.
\end{equation}
we obtain immediately that
\begin{equation}
{H_k(x)\over (-2)^k}
= S_k(\HH -kx).
\end{equation}

On the other hand, the moments of the weight function
\[
w(x) = e^{-x^2}\,, \qquad (x \in \R)
\]
are easily found to be 
\[
{\int_{-\infty}^{+\infty} x^k\,w(x)dx \over
\int_{-\infty}^{+\infty} w(x) dx}
=
\left\{
\matrix{\displaystyle {(2l)!\over 2^{2l}\,l!}
&\mbox{if $k=2l$,}\cr\noalign{\vskip 2mm}
0 & \mbox{if $k=2l+1$,}
}\right.
\,.
\]
This implies, by application of Theorem~\ref{TH1}, that
\begin{equation}\label{HERM}
\det\left[
{H_{n+i+j}(x)\over (-2)^{n+i+j}}
\right]_{0\le i,j \le l-1}
= \
E_{l,n} \,
\Wr(H_l(\sqrt{-1}x),\ldots , H_{l+n-1}(\sqrt{-1}x))\,,
\end{equation}
where $E_{l,n}$ is given by
\[
E_{l,n} = i^{\lfloor{l+n\over 2}\rfloor - \lfloor{l\over 2}\rfloor}
\, 2^{(n+l-1)(l-n)/2}\, {\prod_{k=n}^{l+n-1} k! \over \prod_{k=l}^{l+n-1} k!}
\,.
\]
Note that (\ref{HERM}) is not the formula (18.1) of \cite{KS}.
Indeed, the formula of Karlin and Szeg\"o, which is simpler, 
involves a Wronskian
and a Hankel determinant of the same order $n$, 
and is thus of a different type. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  SECTION 5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Miscellaneous examples}
\label{SECT-MIS}

{\it Example 1.} \ Let us consider the polynomial
\[
d_k(x) = \sum_{i=0}^k (-1)^i {k!\over i!} x^{k-i} \,,
\]
which for $x=1$ gives the number of permutations in $\SG_k$
without fixed point.
We have
\[
d_k(x) = x^k \, S_k(\DD - ku) \,,
\]
where $u=1/x$ and $S_m(\DD)=m!$.
The Hankel determinant $S_{(m^{m+1})}(\DD)$ is easily found
to be
\[
S_{(m^{m+1})}(\DD) = (-1)^{m(m+1)/2} \, \left(
\prod_{k=0}^m k! \right)^2 \,,
\]
and by Theorem~\ref{TH1}, we get
\[
\det\left[
d_{n+i+j}(x)
\right]_{0\le i,j \le l-1}
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\]
\[
\qquad\qquad
=
(-1)^{n(n+1)/2}\,\prod_{k=1}^l k!\, \prod_{k=n}^{l+n-1} k!
\ x^{(l+1)(l+n-1)}\,
\Wr(L_l(u),\ldots , L_{l+n-1}(u))\,,
\]
where $u=1/x$ and $L_m(u) = L_m^{(0)}(u)$ is the (ordinary)
Laguerre polynomial.

\bigskip
\noindent
{\it Example 2.} \ 
Let $E_n(x)$ denote the $n$th Euler polynomial defined
via the generating series
\[
\sum_{n\ge 0} E_n(x)\,{t^n\over n!}
= {2e^{tx}\over e^t+1}\,.
\]
Then, we have 
\[
E_n(x)=2^{-n}\,\sum_{k=0}^n E_k\, {n \choose k}\,  (2x-1)^{n-k} \,,
\]
where $E_k = 2^k E_k(1/2)$ is the $k$th Euler number.
(Note that $E_{2l+1} = 0$).
Thus, putting
\[
S_k(\EE) = E_k \,,
\]
we have $E_n(x) = 2^{-n}\,S_n(\EE-nu)$, with $u=1-2x$.

The calculation of the orthogonal polynomials $S_{(n^n)}(\EE-u)$
is known to be equivalent to the determination of the (formal)
continued fraction expansion of the associated power series 
$\sum_{n\ge 0} S_n(\EE) z^n$.
But this continued fraction has been computed by Stieltjes \cite{Sti},
who found that
\begin{equation}\label{STIELT}
\sum_{n\ge 0} E_n u^{-n-1}
=\int_0^\infty {e^{-ut}\over {\rm ch\,}t} dt
={1\over u+{\strut\displaystyle 1\over \displaystyle u+
{\strut\displaystyle 4\over \displaystyle u+
{\strut\displaystyle 9\over \displaystyle u+
{\strut\displaystyle 16\over \displaystyle \strut u^{\strut} +\cdots}}}}}
\end{equation}
It follows that 
\begin{equation}
S_{(n^n)}(\EE-u)
=
(-1)^n\, \left(\prod_{k=1}^{n-1} k!\right)^2\,
\sum_{l=0}^{\lfloor {n\over 2}\rfloor}
\sigma_{l,n} \,u^{n-2l}
\,,
\end{equation}
where 
\begin{equation}
\sigma_{l,n} =
\sum k_1^2\, k_2^2 \cdots k_l^2
\,,
\end{equation}
the sum running over all integer sequences $(k_i)_{1\le i\le l}$
satisfying
\begin{equation}
1\le k_i < n\,,\qquad k_{i+1}-k_i \ge 2 \,.
\end{equation}

Therefore the specialization of Theorem~\ref{TH1} to the sequence of Euler
polynomials reads 
\begin{equation}
\det\left[
E_{n+i+j}(x)
\right]_{0\le i,j \le l-1}
=
F_{l,n} \, \Wr(\pi_l(u),\ldots ,\pi_{l+n-1}(u)) \,,
\end{equation}
where 
\begin{equation}
F_{l,n} = {(-1)^{nl+l(l-1)/2} (1!\,2!\,\cdots (l-1)!)^2\over
2^{l(l+n-1)}\,1!\,2!\,\cdots (n-1)!}\,,
\end{equation}
and 
\begin{equation}
\pi_m(u) = \sum_{k=0}^{\lfloor {m\over 2}\rfloor}
\sigma_{k,m} \,u^{m-2k}
\end{equation}
is the denominator of the $m$th partial fraction of (\ref{STIELT}).

\bigskip
\noindent
{\it Example 3.} \
Let $B_n(x)$ denote the $n$th Bernoulli polynomial defined
via the generating series
\[
\sum_{n\ge 0} B_n(x)\,{t^n\over n!}
= {te^{tx}\over e^t-1}\,.
\]
Then, we have
\[
B_n(x)=\sum_{k=0}^n B_k\, {n \choose k}\, x^{n-k} \,,
\]
where $B_k =  B_k(0)$ is the $k$th Bernoulli number.
(Note that $B_{2l+1} = 0$ for $l\ge 1$).
Thus, putting
\[
S_k(\BB) = B_k \,,
\]
we have $B_n(x) = S_n(\BB-nu)$, with $u=-x$.
 
The calculation of the orthogonal polynomials $S_{(n^n)}(\BB-u)$
is equivalent to the 
continued fraction expansion of
$\sum_{n\ge 0} S_n(\BB) z^n$
which has been obtained by Rogers \cite{Rog}:
\[
\sum_{n\ge 0} B_n u^{-n-1}
=\int_0^\infty {te^{-ut}\over e^t-1} dt
=2\int_0^\infty {ye^{-(1+2u)y}\over {\rm sh\,}y} dy 
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
\]
\begin{equation}\label{ROGERS}
\qquad\qquad
={2\over 1+2u+{\strut\displaystyle 1^{\strut 4}\over \displaystyle 3(1+2u) +
{\strut\displaystyle 2^{\strut 4}\over \displaystyle 5(1+2u) + 
{\strut\displaystyle 3^{\strut 4}\over \displaystyle 7(1+2u) + 
{\strut\displaystyle 4^{\strut 4}\over \displaystyle \strut 9(1+2u)^{\strut} +\cdots}}}}}
\end{equation}
It follows that
\begin{equation}
S_{(n^n)}(\BB-u)
=
(-1)^n\, S_{((n-1)^n)}(\BB)\,\varphi_n(u+1/2) \,
\end{equation}
where
\begin{equation}
 S_{((n-1)^n)}(\BB) = {1\over 4^{{n\choose 2}}\,(2n-1)!!}\,
\prod_{k=2}^{n-1} {k!^4\over (2k-1)!!^2}\,,\qquad
\varphi_n(w) = 
\sum_{l=0}^{\lfloor {n\over 2}\rfloor}
\tau_{l,n} \,w^{n-2l}
\,,
\end{equation}
and 
\begin{equation}
\tau_{l,n} =
{1\over 4^l}\,\sum {k_1^4\over (2k_1-1)(2k_1+1)}\, \cdots 
{k_l^4 \over (2k_l-1)(2k_l+1)}
\,,
\end{equation}
the sum running over all integer sequences $(k_i)_{1\le i\le l}$
satisfying
\begin{equation}
1\le k_i < n\,,\qquad k_{i+1}-k_i \ge 2 \,.
\end{equation}

Therefore the specialization of Theorem~\ref{TH1} to the sequence of 
Bernoulli polynomials reads
\begin{equation}
\det\left[
B_{n+i+j}(x)
\right]_{0\le i,j \le l-1}
=
G_{l,n} \, 
\Wr\left(\varphi_l\left({1\over 2}-x\right),\,\ldots 
,\,\varphi_{l+n-1}\left({1\over 2}-x\right)\right) \,,
\end{equation}
where
\begin{equation}
G_{l,n} = {(-1)^{nl+l(l-1)/2}\over 
4^{{l\choose 2}}\,
1!\,2!\,\cdots (n-1)!\,(2l-1)!!}
\,\prod_{j=2}^{l-1}
{j!^4 \over (2j-1)!!^2}\,.
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  SECTION 6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A transformation of alphabets}
\label{SECT-OTHER}

There are other formulas in \cite{KS}, Section~28 which suggest an
algebraic approach. 
They can all be deduced from the following general
%
\begin{proposition}\label{DER}
Let $E$ be an arbitrary alphabet. Define for $r\ge l$
\[
\Delta_{l,r}(u) =
\left|
\matrix{
S_l(E-lu) & \cdots & S_{2l-1}(E-(2l-1)u) & S_{r+l}(E-(r+l)u)
\cr\noalign{\vskip 1mm}
S_{l-1}(E-(l-1)u) & \cdots & S_{2l-2}(E-(2l-2)u) & S_{r+l-1}(E-(r+l-1)u)
\cr\noalign{\vskip 1mm}
\vdots & \ddots & \vdots  & \vdots
\cr\noalign{\vskip 1mm}
S_0(E) & \cdots & S_{l-1}(E-(l-1)u)  & S_r(E-ru)
}
\right|
\,.
\]
Then, we have
\begin{equation}\label{REFDER}
\Delta_{l,r}(u) =
{r\choose l}\, S_{(l^{l+1})}(E)\, S_{r-l}(d_lE - (r-l)u)
\,,
\end{equation}
where the alphabet $d_lE$ is defined from the alphabet $E$ by
\[
S_m(d_lE) = {S_{(l^l(l+m))}(E)\over {l+m \choose l} S_{(l^{l+1})}(E)}\,,
\qquad (m\ge 0).
\]
\end{proposition}
The proof of Proposition~\ref{DER} is elementary.
First, by several applications of Lemma~\ref{JACOBI}, one  
can write
\[
\Delta_{l,r}(u) =
S_{(l^lr)}(E,\ldots ,E,E-ru)\,.
\]
Then, one uses (\ref{PHI}) to rewrite this determinant as a
sum and obtain the right-hand side of (\ref{REFDER}).

Taking this into account, the formulas of Section 28 of \cite{KS}
for determinants of the type $\Delta_{l,r}(u)$ whose elements
are orthogonal polynomials of the classes (i) (ii) (iii) of
Theorem~\ref{TH2}, are equivalent to the following
properties of the alphabets $\UU_\lambda$, $\LL_\lambda$
and $\HH$ associated with these polynomials (see  Section~\ref{SECT-SPEC}):
\begin{equation}
d_l \UU_\lambda = \UU_{\lambda + l}\,,\quad
d_l \LL_\alpha = \LL_{\alpha + 2l}\,,\quad
d_l \HH = \HH \,.
\end{equation}
We believe that these properties, which represent nontrivial
evaluations of certain determinants, are to be added to the remarkable
algebraic properties of the classical polynomials discovered
by Burchnall and recalled in Theorem~\ref{TH2}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Acknowledgements}

The results of this paper were obtained in 1991
during the preparation of my doctoral thesis.
I want to express my gratitude to A. Lascoux,
who arose my interest in orthogonal polynomials and
taught me that they should be regarded as special
Schur functions.
I also want to thank S. Milne for convincing me that this
work should be written down because it might find
applications in the vast program on exact sums of squares
formulas that he has recently unveiled in \cite{Mi}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% BIBLIOGRAPHY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip\bigskip
\begin{thebibliography}{ABC} \scriptsize
%
\bibitem{As1} {\sc R. Askey}, {\it On some problems posed by Karlin and 
Szeg\"o concerning orthogonal polynomials},
Boll. Unione Mat. Ital. {\bf 20} (1965), 125-127.
%
\bibitem{As2} {\sc R. Askey}, {\it An inequality for the classical
polynomials}, Nederl. Akad. Wetensch. Proc. Ser. A {\bf 73}
(1970), 22-25.
%
\bibitem{BSS} {\sc E. F. Beckenbach, W. Seidel, O. Sz\'asz},
{\it Recurrent determinant of Legendre and ultraspherical polynomials},
Duke Math. J. {\bf 18} (1951), 1-10.
%
\bibitem{Bu} {\sc J. L. Burchnall}, {\it An algebraic property of the
classical polynomials}, Proc. London Math. Soc. III {\bf 1}
(1951), 232-240.
%
\bibitem{Da1} {\sc A. Danese}, {\it Explicit evaluation of Tur\'an
expressions}, Ann. Mat. Pura. Appl. {\bf 38} (1955), 339-348.
%
\bibitem{Da2} {\sc A. Danese}, {\it Some identities and inequalities 
involving ultraspherical polynomials}, Duke Math. J.
{\bf 26} (1959), 349-359.
%
\bibitem{Fo} {\sc G. E. Forsythe}, {\it Second order determinants
of Legendre polynomials}, Duke Math. J. {\bf 18} (1951), 361-371.
%
\bibitem{Ga1} {\sc G. Gasper}, {\it An inequality of Tur\'an type for 
Jacobi polynomials}, Proc. Amer. Math. Soc. {\bf 32} (1972),
435-439.
%
\bibitem{Ga2} {\sc G. Gasper}, {\it On two conjectures of Askey
concerning normalized Hankel determinants for the classical
polynomials}, SIAM J. Math. Anal. {\bf 4} (1973), 508-513.
%
\bibitem{Il} {\sc L. Iliev}, {\it Laguerre entire functions and 
Tur\'an inequalities}, Studies in Pure Mathematics, 
To the memory of P. Tur\'an, Ed. P. Erd\"os, Birkh\"auser 1983,
347-355.
%
\bibitem{Ja} {\sc C. G. J. Jacobi}, {\it De functionibus
alternantibus earumque divisione per productum
e differentiis elementorum conflatum},
Crelle Journal {\bf 22} (1841), 360-371.
%
\bibitem{KS} {\sc S. Karlin, G. Szeg\"o}, {\it On certain
determinants whose elements are orthogonal polynomials},
Journal d' Analyse Math. {\bf 8} (1961), 1-157.
%
\bibitem{La} {\sc A. Lascoux}, {\it Inversion des matrices de Hankel},
Linear Algebra Appl. {\bf 129} (1990), 77-102.
%
\bibitem{LSH} {\sc A. Lascoux, Shi He}, {\it G\'en\'eralisation 
de la formule de Darboux-Christoffel},
C. R. Acad. Sci. Paris, {\bf 300} (1985), 681-683.
%
\bibitem{Li} {\sc D. E. Littlewood}, {\it The theory of group
characters and matrix representations of groups},
2nd ed. Oxford 1958.
%
\bibitem{Le1} {\sc B. Leclerc}, {\it On identities satisfied
by minors of a matrix}, Adv. Math. {\bf 100} (1993), 101-132.
%
%\bibitem{Le2} {\sc B. Leclerc}, {\it Powers of staircase Schur
%functions and symmetric analogues of Bessel polynomials},
%Discrete Math. {\bf 153} (1996), 213-227.
%
\bibitem{LT} {\sc B. Leclerc, J.-Y. Thibon}, {\it Analogues
sym\'etriques des polyn\^omes de Bessel},
C. R. Acad. Sci. Paris, {\bf 315} (1992), 527-530.
%
\bibitem{Mi} {\sc S. C. Milne}, {\it New infinite families
of exact sums of squares formulas, Jacobi elliptic functions,
and Ramanujan's tau function},
Proc. Natl. Acad. Sci. USA, {\bf 93} (1996), 15004-15008.
%
\bibitem{Mcd} {\sc I. G. Macdonald}, {\it Symmetric functions and 
Hall polynomials}, 2nd edition, Oxford 1995.
%
\bibitem{Pa} {\sc M. L. Patrick}, {\it Some inequalities concerning
Jacobi polynomials}, SIAM J. Math. Anal. {\bf 2} (1971), 213-220.
%
\bibitem{Rog} {\sc L. J. Rogers}, {\it On the representation of certain
asymptotic series as convergent continued fractions},
Proc. London Math. Soc. {\bf 4} (1907), 72-89.
%
\bibitem{Saa} {\sc L. Saalsch\"utz}, {\it Einfache Determinantens\"atze},
Berichte d. phys.-\"okon. Ges. (K\"onigsberg), (1903), 8-9.
%
\bibitem{Sk} {\sc H. Skovgaard}, {\it On inequalities of the Tur\'an type},
Math. Scand. {\bf 2} (1954), 65-73.
%
\bibitem{Sti} {\sc T. J. Stieltjes}, {\it Sur la r\'eduction en fraction
continue d'une s\'erie proc\'edant suivant les puissances des\-cendantes
de la variable}, Ann. Fac. Sci. Toulouse, {\bf 3} (1889), 1-17.
%
\bibitem{Sza1} {\sc O. Sz\'asz}, {\it Inequalities concerning ultraspherical
polynomials and Bessel functions}, Proc. Amer. Math. Soc.
{\bf 1} (1950), 256-267.
%
\bibitem{Sza2} {\sc O. Sz\'asz}, {\it  Identities and inequalities concerning 
orthogonal polynomials and Bessel functions},
Journal d'Analyse Math. {\bf 1} (1951), 116-134.
%
\bibitem{Sze1} {\sc G. Szeg\"o}, {\it On an inequality of Tur\'an concerning
Legendre polynomials}, Bull. Amer. Math. Soc. {\bf 54} (1948),
401-405.
%
\bibitem{Sze2} {\sc G. Szeg\"o}, {\it Orthogonal polynomials},
Amer. Math. Soc. Colloquium Publ. {\bf 23}, Providence, RI, 1975.
%
\bibitem{Tu} {\sc P. Tur\'an}, {\it On the zeros of the polynomials
of Legendre}, \v Casopis pro P\v estov\'ani
Matematiky a Fysiky,
{\bf 75} (1950), 113-122.
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\bibitem{Turn} {\sc H. W. Turnbull}, {\it The theory of determinants,
matrices, and invariants}, Dover 1960.
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\bibitem{VLR} {\sc K. Venkatachaliengar, S. K. Lakshmana Rao},
{\it On Tur\'an's inequality for ultraspherical polynomials},
Proc. Amer. Math. Soc. {\bf 8} (1957), 1075-1087.
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