%%%%%%%%% TIENE 20 PAGINAS COMPILADAS --- PAPER 1 --- %%%%%%%%%
%%%%%%%%% Alvarez-Nodarse, et al.

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\begin{document}
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\markboth{\sc \centerline{Proceedings of the IWOP'96} }{\sc \centerline{Proceedings of the IWOP'96} }


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\input referencia

\begin{center}
  { \bf ON SOME POLYNOMIAL MAPPINGS FOR MEASURES. APPLICATIONS.}
%%{\Large \bf On Some Polynomial Mappings for Measures. Applications.}
\end{center}
\begin{center}
{\bf R.Alvarez-Nodarse, F.Marcell\'an and J.Petronilho}
\end{center}

\begin{abstract}
 {Let $\{P_n\}_{n\geq 0}$ be a given system 
of monic orthogonal polynomials, 
$\pi_k$ and $\theta_{m}$ two fixed monic polynomials 
of degrees $k$ and $m$ ($0\leq m \leq k-1$), 
respectively, and $\{Q_n\}_{n\geq 0 }$ a simple set 
 of monic polynomials such that $Q_{kn+m}(x)=\theta_m(x)P_n(\pi_k(x)),$ 
for all $n=0,1,2,\ldots $. The cases when $T(x)$
 is a polynomial of degree $2$ or $3$ is considered. 
We find necessary and 
sufficient conditions in order that $\{Q_n\}_{n\geq 0 }$
 be a system of orthogonal polynomials and give the relation between the
 moment functionals associated with the  orthogonal systems
$\{P_n\}_{n\geq 0 }$ and $\{Q_n\}_{n\geq 0 }$. 
In particular, we characterize the positive definite case}.
 Some applications  to the solution of the eigenvalue problem for a 
tridiagonal 2 or 3-Toeplitz matrix, as well as to a quantum physics model are given.
\end{abstract}

\begin{description}
\item  {\bf Key words and Phrases}: Orthogonal 
Polynomials, Recurrence coefficients, \\
\phantom{Key words and Phrases}
Polynomial mappings, Toeplitz matrices. 
%\item  {\bf AMS (MSC 1991) Subject Classification}: Primary 42C05. 
\end{description}

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\section{Introduction and preliminaries}

Let us propose the general problem:
\vspace{0.5em}

P. {\it Let $\{P_n\}_{n\geq 0}$ be a given system 
of monic orthogonal polynomials, 
$\pi_k$ and $\theta_{m}$ two fixed monic polynomials 
of degrees $k$ and $m$ ($0\leq m \leq k-1$), 
respectively, and $\{Q_n\}_{n\geq 0 }$ a simple set 
(i.e., {\rm deg} $Q_n$ $=n$ for all $n=0,1,2,\ldots $)
 of monic polynomials such that
$$
Q_{kn+m}(x)=\theta_m(x)P_n(\pi_k(x))  \label{problema}
$$
for all $n=0,1,2,\ldots $. Find necessary and 
sufficient conditions in order that $\{Q_n\}_{n\geq 0 }$
 be a system of orthogonal polynomials.
 Under such conditions, what is the relation between the
 moment functionals associated with the
 orthogonal systems
$\{P_n\}_{n\geq 0 }$ and $\{Q_n\}_{n\geq 0 }$? 
In particular, to characterize the positive definite case}.
\vspace{0.5em}

In this paper we present the solution of this problem 
for the cases $k=2,3$ and for the corresponding
 possible choices of $m$. The proof of the results presented here
can be found in \cite{PacoZe5}, 
\cite{PacoZe6} and \cite{PacoZe3}.
The previous problem was solved by T.S.Chihara
in \cite{Chihara}
for the case $k=2,m=0$, when  $\pi_2(x)\equiv x^2$,
 the initial sequence, 
$\{P_n\}_{n\geq 0}$, is a symmetric MOPS, and 
requiring that $\{Q_n\}_{n\geq 0}$ is also a 
symmetric MOPS (see also \cite{van1}). For $k=3,m=0$, it was 
partially solved in 
\cite{BarrucandDickinson} and \cite{PacoGabriela}.
In fact, in \cite{BarrucandDickinson},
 P.Barrucand and D.Dickinson considered this 
problem under the assumption that 
$\{P_n\}_{n\geq 0}$ is a symmetric orthogonal polynomial 
system and $\pi _3$ a general odd polynomial of
 degree 3, and requiring that 
$\{Q_n\}_{n\geq 0 }$ is also a symmetric orthogonal 
polynomial system; on the other hand, in 
\cite{PacoGabriela}, F.Marcell\'an and
G.Sansigre studied the case $\pi_3(x)\equiv x^3$
 without any restriction on 
$\{P_n\}_{n\geq 0 }$ or $\{Q_n\}_{n\geq 0 }$.
Nevertheless, in these contributions, only sufficient
 conditions were found and the relations 
between the functionals were not studied.
For the positive definite case, a general approach 
to the case $m=0$ (for any $k$) was 
provided by Geronimo and Van Assche 
\cite{GeronimoVanAssche}. In fact, these authors  
proved that given a sequence $\{p_n\}_{n\geq 0 }$
of polynomials orthonormal with respect to some 
positive measure $\mu_0$ supported on the 
bounded interval $[-1,1]$ and a polynomial $T(x)$ of
fixed degree $k\geq2$ with distinct zeros and such
that $|T(y_i)|\geq1$, where $y_i$ ($i=1,2,\ldots,k-1$)
 are the zeros of $T^{\prime}$, then there always exists  
a positive measure $\mu$ and a sequence of 
polynomials $\{q_n\}_{n\geq 0 }$ orthonormal 
with respect to $\mu$ such that 
$q_{kn}(x)=p_n(T(x))$ for all $n=0,1,2,\ldots$. 
We also refer to the contributions by M.H.Ismail \cite{Ismail}, 
M.H.Ismail, J.Charris and S.Monsalve 
\cite{CharrIsmailMons} 
in connection with sieved orthogonal polynomials, 
as well as several papers by  F.Peherstorfer 
\cite{Peherstorfer1}, \cite{Peherstorfer2}, 
related to orthogonality on several intervals and the paper by 
Geronimo and Van Assche \cite{GeronimoVanAssche2} where the orthogonal polynomials with asymptotically 
periodic recurrence coefficients were considered.\\


We recall the definition of an orthogonal
polynomial system (OPS). If {\bf u} is a linear 
functional on $\bkP$ (the space of all polynomials
 with complex coefficients) and 
$\{P_n\}_{n\geq 0 }$ a sequence of polynomials, 
$\{P_n\}_{n\geq 0}$ is said 
to be orthogonal with respect to {\bf u } if
 it is a simple set and $ \langle {\bf u},P_n P_m \rangle =k_n\delta _{nm}$,  
for all $n=0,1,2,\ldots $, where $\{k_n\}_{n\geq 0 }$
is a sequence of nonzero complex numbers. In such a situation the functional {\bf u}
is said to be regular or quasi-definite (cf. \cite[page 16]{Chihara}). 
In this paper we will consider monic orthogonal
polynomial systems (MOPS). All MOPS are characterized
by a three-term recurrence relation of the form
\begin{equation}
\begin{array}{l}
xP_n(x)=P_{n+1}(x)+\beta _n P_n(x)+\gamma _n P_{n-1}(x)\,,
\quad n\geq 1, \esp P_0=1\,,\quad P_1(x)=x-\beta _0,
\end{array}
\label{rrtt}
\end{equation}
where $\{\beta _n\}_{n\geq 0 }$ and 
$\{\gamma _n\}_{n\geq 1}$ are
sequences of complex numbers, with 
$\gamma _n\neq 0$ for all $n\geq 1$. 
As usual, we denote by 
 $\{P_n^{\left( 1\right) }\}_{n\geq 0}$,
$\{P_n^{*}(c;.)\}_{n\geq0}$ (with $c\in\bkC$, and  
$P_n(c)\neq 0$ 
for all $n\geq 0$) and
$\{P_n^\lambda \}_{n\geq 0 }$ ($\lambda\in\bkC$), respectively, 
the sequence 
of the polynomials of the first kind associated with 
$\{P_n\}_{n\geq 0}$ \cite[p.86]{Chihara},  the
sequence of monic kernel polynomials of $K$-parameter
$c$ associated with $\{P_n\}_{n\geq 0}$ \cite[p.35]{Chihara}
and the co-recursive sequence of $\{P_n\}_{n\geq 0 }$ 
for the
modification $\lambda $ \cite{Chihara1}. Explicitely, for the kernel polynomials
we have
\bq
P_n^{*}(c;x)= \frac{1}{x-c} \left[ P_{n+1}(x) - \frac{P_{n+1}(c)}{P_{n}(c)} P_n(x) \right].
\label{pol-ker}
\eq
The associated polynomials are defined by 
\bq
P_n^{\left( 1\right) } = \frac{1}{\langle {\bf u},1 \rangle} 
\left\langle {\bf u}_y,\frac{P_{n+1}(x)-P_{n+1}(y)}{x-y} \right\rangle,
\eq
where ${\bf u}_y$ means that ${\bf u}$ acts on the variable $y$.
Finally, the co-recursive polynomials are given by 
\bq
 P_n^\lambda (x)=P_n(x)-\lambda P_{n-1}^{(1)}(x). 
\eq
The following notation will be used
$$
D_n^{\lambda ,\mu }(x,y)=\frac 1{y-x}\left| 
\begin{array}{cc}
P_n^\lambda (x) & P_{n+1}^\lambda (x) \\ 
P_n^\mu (y) & P_{n+1}^\mu (y)
\end{array}
\right| \quad  {\rm if} \quad x\neq y \, .
$$
 The co-recursive polynomials are important in order
 to establish the regularity
conditions for a linear functional associated with an
 inverse polynomial
modification of a regular (quasi-definite) functional.
In fact,  if ${\bf u}$ is regular, for fixed 
$\lambda ,c\in \bkC$ and with ${\bf u}(\lambda ,c)$ 
defined by the distributional
equation $(x-c){\bf u}(\lambda ,c)=-\lambda {\bf u}$, 
i.e., $\dst {\bf u}(\lambda ,c)=u_0\delta _c-
\lambda (x-c)^{-1}{\bf u\,}$, 
where $\delta _c$ represents the Dirac measure at the 
point $c$ and $(x-c)^{-1}{\bf u}$ 
is the linear functional defined by 
$\langle (x-c)^{-1}{\bf u} , f \rangle =
\langle {\bf u} ,[f(x)-f(c)]/(x-c) \rangle $,  
then ${\bf u}(\lambda ,c)$ is regular if and only 
if $\lambda \neq 0$ and 
$P_n^\lambda (c)\neq 0$ for all $n=0,1,2,\ldots $. 
Under such conditions the
corresponding MOPS, $\{P_n(.;\lambda ,c)\}_{n\geq 0}$ ,
 is given by (see \cite{Maroni})
$$
P_n(x;\lambda ,c)=P_n(x)-
\frac{P_n^\lambda (c)}{P_{n-1}^\lambda (c)}
P_{n-1}(x)\quad ,\quad n\geq 0\,.  
$$
We recall the definition of the Stieltjes formal series,
 $S_{{\bf u}}(z)$, corresponding to a given functional {\bf u}, 
\[
S_{{\bf u}}(z)=-\sum_{n\geq 0}\frac{u_n}{z^{n+1}} \ ,
\]
where $u_n=\langle ${\bf u}$,x^n\rangle $ is the 
moment of order $n$ of ${\bf u}$.\\

In the previous results we have considered 
``formal orthogonality'', in the
sense that the orthogonal polynomials 
$\{P_n\}_{n\geq 0}$ are related to a
numerical sequence 
$u_n=\langle {\bf u},x^n\rangle 
\;,\;n=0,1,2,\ldots $,
ignoring whether these numbers are
 actual moments of some weight or
distribution function on some support or not.
 In order to determine under what
conditions this can occur, we must analyze 
when a regular
linear functional ${\bf u}$ is positive definite,
 i.e., $\langle {\bf u},f\rangle >0$ 
for all $f\in \bkP$ 
such that $f(x)\geq0\,,\,\forall x\in \bkR$ 
and $f\not \equiv 0$. In fact, a
sequence of orthogonal polynomials 
$\{P_n\}_{n\geq 0}$ with respect to some
linear functional ${\bf u}$ is said to be
 orthogonal in the positive
definite sense if ${\bf u}$ is positive definite.
 By a representation
theorem \cite[Chapter II]{Chihara}, ${\bf u}$ is
positive definite if and only if it admits
 an integral representation, in
terms of the Stieltjes integral, of the form $\langle {\bf u},\pi \rangle =
\int_{-\infty }^{+\infty }\pi(x)d\sigma (x)$, 
for every polynomial $\pi$ , where 
$\sigma $ is a distribution function, 
i.e., a function $\sigma :\bkR\rightarrow \bkR$
which is nondecreasing, has infinitely many 
points of increase 
and such that all the moments 
$\int_{-\infty }^{+\infty }x^{2n}d\sigma (x)$ 
($n=0,1,2,\ldots$) are finite. 
For a given distribution function $\sigma $,
 we denote the corresponding
Stieltjes function by $F(.;\sigma )$,
\bq
F(z;\sigma )=\int_{-\infty }^{+\infty }
\frac{d\sigma (t)}{t-z}\, .
\label{stieltjes}
\eq 

The structure of the paper is the following. 
In Section 2 we give the
characterizations for the above problem in 
the cases $k=2,m=0,1$ and  $k=3,m=0,2$. The
solution for the case $k=3,m=1$ is very similar
and  we omit it.
As an application 
of our results, in Section 3 we give the
solution for the eigenproblem for a 
tridiagonal 2 and 3-Toeplitz matrix. We mention that
the problem for a tridiagonal
2-Toeplitz matrix has been solved by M.J.Gover
\cite{Gover} (see also \cite{PacoZe5}). Finally, in Section 4
we analyze a particular case of the so called {\em Chain Model} \cite{haydock}
which is an exactly solvable quantum model.

\section{Results.}

\begin{theorem} \label{solq1}(case $k=2,m=0$)
Let $\{P_n\}_{n\geq 0}$ be a system of monic 
orthogonal polynomials and 
$\{Q_n\}_{n\geq 0}$ a simple set of monic
 polynomials such that 
$$
Q_1(x)=x-b \,,\quad Q_{2n}(x)=P_n(\pi_2(x))
\,,\quad n=0,1,2,\ldots  
$$
where $\pi_2(x)$ is a (monic) polynomial of 
degree $2$ and $b\in \bkC $. Without 
loss of generality, 
write $\pi_2(x)=(x-a)(x-b)+c$. Then, 
$\{Q_n\}_{n\geq 0}$ is a MOPS if
and only if 
$$
 P_n(c)\neq 0\,,
\quad Q_{2n+1}(x)=(x-b)P_n^*(c;\pi_2(x)) 
\, ,\quad n=0,1,2,\ldots \, .
$$
Under such conditions,  if $\{P_n\}_{n\geq 0}$ 
satisfies
 the three-term
recurrence relation {\rm (\ref{rrtt})}, then
 the coefficients $\tilde \beta_n$ 
and $\tilde \gamma _n$, for the 
corresponding three-term 
recurrence relation satisfied by 
$\{Q_n\}_{n\geq 0}$, are 
given by
$$
\tilde \beta _{2n}=b \,,
\quad \tilde \beta _{2n+1}=a \, , \quad
\tilde \gamma _{2n+1}=-
\frac{P_{n+1}(c)}{P_{n}(c)}  \,, \quad
\tilde \gamma _{2n+2}=
-\gamma_{n+1}\frac{P_{n}(c)}{P_{n+1}(c)} 
$$
for all $n=0,1,2,\ldots$. Moreover, 
if $\{P_n\}_{n\geq 0}$
is a MOPS with respect to the linear 
functional ${\bf u}$, then 
$\{Q_n\}_{n\geq 0}$
is a MOPS with respect to the linear 
functional ${\bf v}$ defined on the basis
$\{ \pi_2^n,(x-b)\pi_2^n  \}_{n\geq0}$ of $\bkP$ by
$$
\langle {\bf v},\pi_2^n(x)\rangle =
\langle {\bf u},x^n\rangle
\,,\quad 
\langle  {\bf v}, (x-b)\pi_2^n(x)\rangle =0\, , 
\quad n=0,1,2,\ldots \, .
$$
\end{theorem}
\begin{corollary}\label{associadosq1}
Under the hypothesis of Theorem {\rm \ref{solq1}} the following expressions hold
$$
Q_{2n-1}^{(1)}(x)=(x-a)P_{n-1}^{(1)}(\pi_2(x)) \, , 
\quad  n=0,1,2,\ldots \,  \esp S_{{\bf v}}(z)=(z-a)S_{{\bf u}}(\pi_2(z))\, .
$$
\end{corollary}  
\begin{corollary}\label{solq1+}
Under the hypothesis of Theorem {\rm \ref{solq1}}, 
assume that
$b$ is a real number, $\pi_2$ a real
 polynomial and
 $\left\{ P_n\right\} _{n\geq 0 }$ a MOPS 
in the positive definite sense,
orthogonal with respect to some uniquely determined
distribution function $\sigma$ and 
let $\left[ \xi ,\eta \right] $ be
the corresponding true interval of orthogonality \cite{Chihara}.
Then, $\left\{ Q_n\right\} _{n\geq 0 }$ is
a MOPS with respect to a positive definite 
linear functional if and only if  
$$
 c\leq\ \xi \, , \quad   
Q_{2n+1}(x)=(x-b)P_n^*(c;\pi_2(x))
$$
hold for all $n=0,1,2,\ldots$. Under these
conditions, $\left\{ Q_n\right\} _{n\geq 0 }$ is
orthogonal with respect to the uniquely 
determined distribution function 
$\widetilde{\sigma }$ defined by 
$$
{\rm d}\widetilde{\sigma }(x)=| x-a |
 \frac{{\rm d}\sigma(\pi_2(x))}{\pi_2^{\prime}(x)}
\quad  , \quad 
r\leq |x-{\scriptstyle{\frac{a+b}{2}}} | 
\leq s \, {,}  
$$
where $r=(\xi+\Delta /4)^{1/2}$ and 
$s=(\eta+\Delta /4)^{1/2}$, with 
$\Delta=(b-a)^2-4c$.
\end{corollary}
\begin{theorem}  \label{solq2} (case $k=2,m=1$)
Let $\{P_n\}_{n\geq 0}$ be a system of monic 
orthogonal polynomials and 
$\{Q_n\}_{n\geq 0}$ a simple set of monic
 polynomials such that 
$$
Q_2(a)=\lambda \,,\quad Q_{2n+1}(x)=(x-a)P_n(\pi_2(x))
\,,\quad n=0,1,2,\ldots  
$$
where $\pi_2(x)$ is a (monic) polynomial of 
degree $2$ and $a,\lambda\in \bkC $. Without 
loss of generality, 
write $\pi_2(x)=(x-a)(x-b)+c$. Then, 
$\{Q_n\}_{n\geq 0}$ is a MOPS if
and only if 
$$
\lambda\neq0 \, , \quad P_n^{\lambda}(c)\neq 0\,,
\quad Q_{2n}(x)=P_n(\pi_2(x);\lambda,c) 
\, ,\quad n=0,1,2,\ldots \, .
$$
Under such conditions,  if $\{P_n\}_{n\geq 0}$ satisfies
 the three-term
recurrence relation {\rm (\ref{rrtt})}, then
 the coefficients $\tilde \beta_n$ 
and $\tilde \gamma _n$ for the corresponding three-term 
recurrence relation satisfied by $\{Q_n\}_{n\geq 0}$ are 
given by
$$
\tilde \beta _{2n}=a \,,
\quad \tilde \beta _{2n+1}=b \, , \quad
\quad n=0,1,2,\ldots \, ,
$$
$$
\tilde \gamma_{1}=-\lambda \, , \quad
\tilde \gamma_{2n}=-
\frac{P_{n}^{\lambda}(c)}{P_{n-1}^{\lambda}(c)}  
\,, \quad
\tilde \gamma _{2n+1}=
-\gamma_{n}
\frac{P_{n-1}^{\lambda}(c)}{P_{n}^{\lambda}(c)} 
\, , \quad n=1,2,\ldots \, .
$$
Moreover, if $\{P_n\}_{n\geq 0}$
is a MOPS with respect to the linear 
functional ${\bf u}$, then 
$\{Q_n\}_{n\geq 0}$
is a MOPS with respect to the linear 
functional ${\bf v}$ defined by
$$
\langle {\bf v},\pi_2^n(x)\rangle =
\langle {\bf u}(\lambda,c),x^n\rangle
\,,\quad 
\langle  {\bf v}, (x-a)\pi_2^n(x)\rangle =0
\, , \quad n=0,1,2\ldots \, .
$$
\end{theorem} 
\begin{corollary} \label{associadosq2}
Under the hypothesis of Theorem {\rm \ref{solq1}}, the following expressions hold
$$
Q_{2n-1}^{(1)}(x)=(x-b)P_{n-1}{}^{(1)}(\pi_2(x);\lambda,c) \, , 
\quad  n=1,2,\ldots \, ,\esp 
S_{{\bf v}}(z)=(z-b)S_{{\bf u(\lambda,c)}}(\pi_2(z)).
$$
\end{corollary}
\begin{corollary}\label{solq2+}
Under the hypothesis of Theorem {\rm \ref{solq2}}, 
assume that $a,\lambda$ are real numbers, 
$\pi_2$ a real polynomial and
 $\left\{ P_n\right\} _{n\geq 0 }$ a MOPS 
in the positive definite sense,
orthogonal with respect to some uniquely determined
distribution function $\sigma $ and 
let $\left[ \xi ,\eta \right] $ be
the corresponding true interval of orthogonality \cite{Chihara}.
Further, assume either that $c\leq \xi +\lambda $, or
\bq
c\leq \xi  \, ,  \quad
-\infty < \lim_{n\rightarrow +\infty}
\frac{P_n(\xi)}{P_{n-1}^{(1)}(\xi)}
\equiv A \leq \lambda \leq B \equiv
\lim_{n\rightarrow +\infty}
\frac{P_n(\eta)}{P_{n-1}^{(1)}(\eta)}
\label{condq2}
\eq
(with $B\equiv \infty$ if $\eta=+\infty$) holds. 
Then, $\left\{ Q_n\right\} _{n\geq 0 }$ is
a MOPS with respect to a positive definite 
linear functional if and only if  $ \lambda < 0$ and   
$Q_{2n}(x)=P_n(\pi_2(x);\lambda,c)$, holds for all $n=0,1,2,\ldots$. 
Under these conditions, $\left\{ Q_n\right\} _{n\geq 0 }$ is
orthogonal with respect to the uniquely 
determined distribution function 
$\widetilde{\sigma }$ given by 
$$
{\rm d}\widetilde{\sigma }(x)=M\delta_a(x)-
\frac{\lambda}{| x-a |}
 \frac{{\rm d}\sigma(\pi_2(x))}{\pi_2^{\prime}(x)}
\quad  , \quad 
M:=u_0+\lambda F(c;\sigma)\geq0 \, {,}  
$$
whith either ${\rm \supp}({\rm d}\sigma )\subset \pi_2^{-1}([\xi + \lambda, \eta])\cup \{a\}$ 
if $c\leq \xi +\lambda $, or ${\rm \supp}({\rm d}\sigma)\subset \pi_2^{-1}([\xi, \eta])\cup \{a\}$ 
if {\rm (\ref{condq2})} holds.
\end{corollary} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{solc1} (case $k=3,m=0$)
Let $\{P_n\}_{n\geq 0}$ be a system of monic 
orthogonal polynomials and 
$\{Q_n\}_{n\geq 0}$ a simple set of monic
 polynomials such that 
\begin{equation}
Q_1(0)=-\beta \,,\quad Q_2(\beta )=-\gamma \,,
\quad Q_{3n}(x)=P_n(\pi_3(x))\,,\quad n=0,1,2,\ldots  
\label{p1.1}
\end{equation}
where $\pi_3(x)$ is a (monic) polynomial of 
degree $3$ and $\beta ,\gamma \in \bkC $.
Consider the polynomial 
$$
\rho (x)=\gamma +[\pi _3(x)-\pi_3(\beta )]/(x-\beta) \, .
$$
Let $a_1$ and $a_2$ be the zeros of $\rho $ and put 
$ c_1=\pi_3(a_1)$, $ c_2=\pi _3(a_2)$.
Then, for a fixed pair $(\beta ,\gamma)$, 
$\{Q_n\}_{n\geq 0}$ is a MOPS if
and only if 
$$
\gamma \neq 0\,,\quad P_n(c_1)\neq 0\,,
\quad P_n(c_2)\neq 0\,,\quad
P_n^{*}(c_1;c_2)\neq 0\,,  
$$
$$
Q_{3n+1}(x)={\frac 1{\rho (x)}}
\left[ {P_{n+1}(\pi _3(x))+\gamma  
\frac{P_n^{*}(c_1;c_2)}{P_n(c_2)}\left( x-a_2-
\frac 1\gamma  
\frac{P_{n+1}(c_2)}{P_n^{*}(c_1;c_2)}\right) \,
P_n(\pi _3(x))}\right] {\,},  
$$
$$
Q_{3n+2}(x)={\frac 1{\rho (x)}}
\left[ {\left( x-a_1+\frac 1\gamma  
\frac{P_{n+1}(c_2)}{P_n^{*}(c_1;c_2)}\right) 
P_{n+1}(\pi_3(x))-
\frac{1}{\gamma} 
\frac{P_{n+1}(c_1)P_{n+1}(c_2)}
{P_n(c_1)P_n^{*}(c_1;c_2)}\,
P_n(\pi_3(x))}
\right]   
$$
hold for all $n=0,1,2,\ldots $. In such conditions, if $\{P_n\}_{n\geq 0}$
 satisfies the three-term
recurrence relation {\rm (\ref{rrtt})}, then
 the coefficients $\tilde \beta_n$ 
and $\tilde \gamma _n$ for the corresponding 
three-term 
recurrence relation satisfied by 
$\{Q_n\}_{n\geq 0}$ are 
given by
$$
\tilde \beta _{3n}=\beta \,,
\quad \tilde \beta _{3n+1}=a_1-\frac 1\gamma 
\frac{P_{n+1}(c_2)}{P_n^{*}(c_1;c_2)}\,,
\quad \tilde \beta _{3n+2}=a_2+
\frac 1\gamma \frac{P_{n+1}(c_2)}
{P_n^{*}(c_1;c_2)}\,,  
$$
$$
\tilde \gamma _{3n}=-\gamma \,\gamma _n\,
\frac{P_{n-1}(c_1)P_{n-1}^{*}(c_1;c_2)}
{P_n(c_1)P_n(c_2)}\,,  \quad
\tilde \gamma _{3n+1}=\gamma \,
\frac{P_n^{*}(c_1;c_2)}{P_n(c_2)}\,,\quad
\tilde \gamma _{3n+2}=-\frac 1{\gamma ^2}
\frac{P_{n+1}(c_1)P_{n+1}(c_2)}
{P_n^{*}(c_1;c_2)P_n^{*}(c_2;c_1)}\,.  
$$
Finally, denote by {\bf u} the moment functional 
such that $\{P_n\}_{n\geq0}$
is the corresponding MOPS. Let $\{A_n\}_{n\geq 0}$
 be the basis of $\bkP$ defined by 
$$
A_{3n}(x)=\pi _3^n(x)\,,\quad A_{3n+1}(x)=(x-\beta )
\pi_3^n(x)\,,\quad
A_{3n+2}(x)=(x^2-\beta x-\gamma )\pi _3^n(x)  
$$
for $n=0,1,2,\ldots $. Then, under the above conditions,
 $\{Q_n\}_{n\geq 0}$
is a MOPS associated with the moment linear functional 
{\bf v} defined on
the basis $\{A_n\}_{n\geq 0}$ by 
$$
\langle  {{\bf v}},A_{3n}(x)\rangle =
\langle {\bf u},x^n\rangle
\,,\quad \langle  {{\bf v}},A_{3n+1}(x)\rangle =0\,,
\quad \langle {\bf v},A_{3n+2}(x)\rangle =0, \esp n=0,1,2,\ldots \,\, .
$$
\end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{associadosc1}
Under the hypothesis of Theorem {\rm \ref{solc1}} the following expression hold
$$
Q_{3n+2}^{(1)}(x)=\rho (x)P_n{}^{(1)}(\pi _3(x)), \quad  n=0,1,2,\ldots \, ,
\esp S_{{\bf v}}(z)=\rho(z)S_{{\bf u}}(\pi_3(z))\, {.}
$$
\end{corollary}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{solc1+}
Under the hypothesis of Theorem 
{\rm \ref{solc1}}, assume further that
$\beta, \gamma$ are real numbers, 
$\pi_3$ a real polynomial and
 $\left\{ P_n\right\} _{n\geq 0 }$ 
a MOPS in the positive definite sense,
orthogonal with respect to some
distribution function $\sigma $ 
and let $\left[ \xi ,\eta \right] $ be
the corresponding true interval of orthogonality \cite{Chihara} of 
$\left\{ P_n\right\} _{n\geq 0 } $.
Then, $\left\{ Q_n\right\} _{n\geq 0 }$ is
a MOPS with respect to a positive definite 
linear functional if and only if 
$a_1$ and $a_2$ are real (we assume, 
without loss of generality, that 
$a_1\leq a_2$) and the following two 
conditions hold 
$$\ba{l}
{{\rm (i)}}\,\quad \gamma >0\,\quad  {,}
\quad c_2\leq \xi \quad 
 {,}\quad \eta \leq c_1\,{\rm ;} \\
 \\
{\rm (ii)} \quad Q_{3n+1}(x) \mbox{ and 
$Q_{3n+2}(x)$ are defined as in Theorem \ref{solc1}  for $n=0,1,2,\ldots $}.
\ea
$$
Under these conditions, 
$\left\{ Q_n\right\} _{n\geq 0 }$ is
orthogonal with respect to the uniquely
 determined distribution function 
$\widetilde{\sigma }$ given by 
$$
{\rm d}\widetilde{\sigma }(x)=
\left| \rho (x)\right| \frac{{\rm d}\sigma
(\pi _3(x))}{\pi _3^{\prime }(x)}\quad 
 {,}\quad x\in \pi _3^{-1}\left(
\left[ \xi ,\eta \right] \right) \, {.}  
$$
\end{corollary}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}\label{remar}  
Conditions {\rm (i)} imply that in order to guarantee the existence of
sequences $\left\{ Q_n\right\} _{n\geq 0 }$, satisfying 
{\rm (\ref{p1.1})} and orthogonal in the 
positive definite sense, 
we must start with
sequences $\left\{ P_n\right\} _{n\geq 0 }$ (orthogonal 
in the
positive definite sense) such that the corresponding 
true interval of
orthogonality, $\left[ \xi ,\eta \right] $, is bounded, 
and such that the
condition $c_1\neq c_2$ holds (or, equivalently, 
$a_1\neq a_2$). 
\end{remark}
\begin{remark}
The results of the corollaries \ref{solq1+} and \ref{solc1+} 
agree with  those of Geronimo and
 Van Assche \cite{GeronimoVanAssche}, for the 
particular cases $k=2,3$. Notice that, in
 \cite{GeronimoVanAssche} the authors imposed, 
a priori, the
restrictions ``supp$({\rm d}\sigma )$ compact'' and 
``$\left| \pi_k(y_i)\right| \geq 1$ on the zeros 
$y_i$ of $\pi_k^{\prime }$ ($i=1,2$)'',
which in our case correspond to the condition 
$\pi_k(y_i)\notin ]\xi ,\eta[$. 
For the case $k=3$, it follows from 
Remark \ref{remar} that the condition 
``supp$({\rm d}\sigma)$ 
compact'' is
necessary for the orthogonality of $\{ Q_n\} _{n\geq 0}$.
Furthermore (also for $k=3$), assuming 
(without loss of generality) 
that $y_1<y_2$, since at
these points $\pi _3(y)$ attains its extremal values, 
we have 
$\pi_3(y_1)\leq \pi _3(a_2)=c_2\leq \xi <
\eta \leq c_1=\pi _3(a_1)\leq \pi_3(y_2)$, hence again 
$\pi _3(y_i)\notin ]\xi ,\eta [$ must hold necessarily
for the orthogonality of $\left\{ Q_n\right\}_{n\geq 0}$.
\end{remark}
\begin{remark}
In \cite{PacoGabriela} it was pointed out that
``there does not exist a pair of orthogonal 
polynomial sets $\{ P_n \}_{n\geq0}$ and
$\{ Q_n \}_{n\geq0}$ associated with positive
definite symmetric functionals such that
$Q_{3n}(x)=P_n(x^3)$''. We can generalize  
this observation, removing the restriction 
about the symmetry of the functionals, since
for any choice of the pair $(\beta,\gamma)$,
with $\gamma >0$, the polynomial
$\rho(x)=\gamma+(x^3-\beta^3)/(x-\beta)
\equiv x^2+\beta x+ \beta^2+\gamma$
has no real zeros.
\end{remark}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{theorem}\label{solc3} (case $k=3,m=2$)
Let $\{P_n\}_{n\geq 0 }$be a MOPS and 
$\{Q_n\}_{n\geq 0}$ a simple set of monic polynomials such that 
$$
Q_{3n+2}(x)=(x-a)(x-b)P_n(\pi _3(x))\,,
\quad n=0,1,2,\ldots  
$$
where $a$ and $b$ are fixed complex numbers 
and $\pi _3(x)$ is a monic
polynomial of degree 3. Without loss of generality, write 
\[
Q_1(x)=x-\alpha \,,\quad xQ_2(x)=Q_3(x)+
\beta Q_2(x)+\gamma Q_1(x) 
\]
and put $c=\pi _3(a)$, $d=\pi _3(b)$,
$\lambda =-\gamma (a-\alpha)$ and 
$\mu =-\gamma (b-\alpha )$. 
Assume that $c\neq d$. Then, for a fixed pair 
$\left( \alpha ,\gamma
\right) $, $\{Q_n\}_{n\geq 0 }$ is a MOPS if and only if 
\[
\gamma \neq 0\,,\quad \beta =-
\left[ a+b+\pi _3^{\prime \prime }(0)/2\right],\esp 
P_n^\lambda (c)\neq 0\,,
\quad P_n^\mu (d)\neq 0\,,\quad 
D_n^{\lambda ,\mu}(c,d)\neq 0\,, 
\]
\[
Q_{3n+3}(x)=P_{n+1}(\pi _3(x))+
\frac{d_n^{\lambda ,\mu }(c,d)}{P_n^\lambda
(c)P_n^\mu (d)}\left( x-a-
\frac{P_{n+1}^\lambda (c)P_n^\mu (d)}{d_n^{\lambda
,\mu }(c,d)}\right) P_n(\pi _3(x))\,, 
\]
\[
Q_{3n+4}(x)=\left( x-b+\frac{P_{n+1}^\lambda (c)
P_n^\mu (d)}{d_n^{\lambda
,\mu }(c,d)}\right) P_{n+1}(\pi _3(x))-
\frac{P_{n+1}^\lambda (c)P_{n+1}^\mu
(d)}{d_n^{\lambda ,\mu }(c,d)}P_n(\pi _3(x)) 
\]
hold for all $n=0,1,2,\ldots $ where $\dst d_n^{\lambda ,\mu }(c,d)=\frac{d-c}{a-b}
D_n^{\lambda ,\mu }(c,d)$. Under such conditions, if $\{P_n\}_{n\geq 0 }$
satisfies the three-term
recurrence relation (\ref{rrtt}), then the coefficients
 $\widetilde{\beta}_n $ 
and $\widetilde{\gamma }_n$ for the 
corresponding three-term recurrence
relation satisfied by $\{Q_n\}_{n\geq 0 }$  are given by 
\[
\widetilde{\beta }_0=\alpha \,,\quad
\widetilde{\beta }_1=a+b-\alpha
\,,
\]
\[
\widetilde{\beta }_{3n+2}=\beta \,,\quad
\widetilde{\beta }_{3n+3}=b-
\frac{P_{n+1}^\lambda (c)
P_n^\mu (d)}{d_n^{\lambda ,\mu }(c,d)}\,,\quad 
\widetilde{\beta }_{3n+4}=a+
\frac{P_{n+1}^\lambda (c)P_n^\mu (d)}
{d_n^{\lambda ,\mu }(c,d)}\,,
\quad n=0,1,2,...
\]
\[
\widetilde{\gamma }_1=-\lambda \mu /
\gamma ^2\,,\quad \widetilde{\gamma }_2=\gamma 
\]
\[
\widetilde{\gamma }_{3n}=
\frac{d_{n-1}^{\lambda ,\mu }(c,d)}{P_{n-1}^\lambda
(c)P_{n-1}^\mu (d)}\,,\quad
 \widetilde{\gamma }_{3n+2}=-\gamma _n
\frac{d_{n-1}^{\lambda ,\mu }(c,d)}
{P_n^\lambda (c)P_n^\mu (d)}\,,
\quad \widetilde{\gamma }_{3n+1}=-
\frac{P_{n-1}^\lambda (c)P_{n-1}^\mu
(d)P_n^\lambda (c)P_n^\mu (d)}
{\left( d_{n-1}^{\lambda ,\mu }(c,d)\right) ^2},
\]
for $n=0,1,2,...$ Furthermore, assume that $\{P_n\}_{n\geq 0}$ is a
 MOPS associated with the
moment functional {\bf u}. Consider the basis 
$\{C_n\}_{n\geq 0}$ of $\bkP$
 such that 
\[
\begin{array}{l}
C_0(x)=1\,,\quad C_{3n+1}(x)=(x-b)\pi_3^n(x)\,, \\
C_{3n+2}(x)=(x-a)(x-b)\pi _3^n(x)\,, 
\quad C_{3n+3}(x)=(x-a)(x-b)^2\pi
_3^n(x),
\end{array}
\]
for $n=0,1,2,\ldots $. Then, under the conditions above,
 $\{Q_n\}_{n\geq 0}$
is a MOPS associated with the linear functional {\bf v}
 defined by ($n=0,1,2,\ldots $)
$$
\begin{array}{l}
\langle  {{\bf v}},C_0(x)\rangle =u_0\,,
\quad \langle  {{\bf v}},C_{3n+1}(x)\rangle 
=\frac \mu \gamma 
\langle {\bf u(}\lambda ,c{\bf )},x^n\rangle \,,  
\langle {\bf v},C_{3n+2}(x)\rangle =0\,,
\quad \langle  {{\bf v}},C_{3n+3}(x)\rangle =0.
\end{array}
$$
\end{theorem}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{corollary}\label{associadosc3}
Under the hypothesis of Theorem {\rm \ref{solc3}}, the following expressions hold
$$\ba{l}
\dst Q_{3n+1}^{(1)}(x)=\left( x-a-b+\alpha \right)
 P_n(\pi _3(x))-\frac{\lambda
\mu }\gamma \,P_{n-1}^{(1)}(\pi _3(x)) ,\esp n=0,1,2,\ldots\\
\\
\dst S_{ {{\bf v}}}(z)=-\frac{u_0(z-a-b+\alpha )+
(\lambda \mu /\gamma )
S_{{\bf u}}(\pi_3(z))}{(z-a)(z-b)}\, {.}  
\ea
$$
\end{corollary}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{corollary}\label{solc3+}
Under the hypothesis of Theorem {\rm \ref{solc3}}, 
assume further that
$\alpha, \gamma, a,b$ are real numbers, 
$\pi_3$ a real polynomial and
 $\left\{ P_n\right\} _{n\geq 0 }$ a MOPS 
in the positive definite sense,
orthogonal with respect to some
distribution function $\sigma$, and 
let $\left[ \xi ,\eta \right] $ be
the corresponding true interval of orthogonality \cite{Chihara}, 
which we assume to be bounded.
Without loss of generality,
suppose that $a>b$ and assume that one of the
two sets of conditions hold, either $c\leq \xi +\lambda \, {,}\quad \eta 
+\mu \leq d\, $, or
\bq
c\leq \xi  \, ,  \quad 
d\geq \eta  \, , \quad
-\infty < \lim_{n\rightarrow +\infty}
\frac{P_n(\xi)}{P_{n-1}^{(1)}(\xi)}
\equiv A \leq \lambda , \mu \leq B \equiv
\lim_{n\rightarrow +\infty}
\frac{P_n(\eta)}{P_{n-1}^{(1)}(\eta)} \, .
\label{condc3}
\eq
Then, for each fixed pair $(\alpha ,\gamma )$,
 $\left\{ Q_n\right\} _{n\geq 0}$ is 
a MOPS in the positive definite sense 
if and only if the
following two conditions hold 
\[
\ba{l}
 {\rm (i)}\,\quad \gamma >0\,\quad  {,} \quad \lambda <0<\mu \quad 
 {,}\quad \beta =
-[a+b+\pi_3^{\prime \prime }(0)/2]\,\,{\rm ;} \\
{\rm (ii)}\mbox{ $\quad Q_{3n+3}(x)$ and $Q_{3n+4}(x)$ 
are defined as in Theorem {\rm \ref{solc3}} for 
$n=0,1,2,\ldots $} .
\ea
\]

\noindent Under these conditions, 
$\left\{ Q_n\right\} _{n\geq 0 }$ is
orthogonal with respect to a uniquely
 determined distribution function 
$\widetilde{\sigma }\ $ such that 
$$
d\widetilde{\sigma }(x)=M\delta _a(x)+
N\delta_b(x)-\frac{\lambda \mu
/\gamma }{\left| (x-a)(x-b)\right| }
\frac{d\sigma (\pi _3(x))}{\pi_3^{\prime}(x)}
\quad \, {,}  
$$
where
$$
M=\frac{\mu /\gamma }{a-b}
\left[ u_0+\lambda F(c;\sigma )\right] \geq 0\,
 {,\quad }
N=-\frac{\lambda /
\gamma }{a-b}\left[ u_0+\mu F(d;\sigma
)\right] \geq 0\, {.} 
$$
Furthermore, either ${\rm \supp}({\rm d}\sigma )\subset 
\pi_2^{-1}([\xi + \lambda, \eta + \mu])\cup \{a,b\}$ 
if $c\leq \xi +\lambda \, {,}\quad \eta 
+\mu \leq d\, $, or ${\rm \supp}({\rm d}\sigma)\subset \pi_2^{-1}([\xi, \eta])\cup \{a,b\}$ 
if {\rm (\ref{condc3})} holds.
\end{corollary}

\begin{remark}
In the previous Corollary, we can assume, 
without loss of generality, that $a>b$
(not $a\geq b$) because if $a=b$ 
then $Q_{3n+2}$ would have a double
zero and so $\left\{ Q_n\right\} _{n\geq 0 }$ 
couldn't be orthogonal
in the positive definite sense.
\end{remark}

\begin{remark}
As a consequence of Corollaries \ref{associadosq1}, 
\ref{associadosq2}, \ref{associadosc1} and
\ref{associadosc3},
 we can state that if 
$\{P_n\}_{n\geq 0}$ is a
semiclassical MOPS (in any of the problems), 
then  $\{Q_n\}_{n\geq 0}$ is also a semiclassical MOPS
(for information about this class of orthogonal 
polynomials, see, e.g., \cite{Maroni1}).
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Applications}
\esp  First of all, let us point out some  
 well known results of the general theory of orthogonal polynomials. 
  As we said before, the orthogonal polynomial sequence (OPS) $\{P_n\}_{n\geq0}$  
   can be characterized   by the three term recurrence relation  (\ref{rrtt}) 
   satisfied by  the polynomials $P_n(x)$ (which are not necessarily monic) \cite{Chihara} 
 $$ 
   xP_n(x)=\alpha_n P_{n+1}(x)+\beta_n P_n(x)+\gamma_nP_{n-1}(x) \quad , \quad n=1,2,... \quad  
 $$ 
   with initial conditions $P_{-1}=0$ and $P_0=1$ (we take this last as a
   normalization condition) where $\{\alpha_n\}_{n\geq0}$, $\{\beta_n\}_{n\geq0}$ and 
   $\{\gamma_n\}_{n\geq1}$ are sequences of complex numbers such that $\alpha_n \gamma_{n+1} \neq0$ for $n\geq0$.  
   In matrix form, this can be written as \esp 
   $x\, \mbox{{\bf P}}_n= J_{n+1}\mbox{{\bf P}}_n+\alpha_n P_{n+1}(x) \mbox{{\bf e}}_n$,
   where $J_{n+1}$ is a tridiagonal matrix of order $n+1$, $\mbox{{\bf P}}_n$ is the 
   polynomial vector $[P_0(x),...,P_n(x)]^T$ and $\mbox{{\bf e}}_n$ is the vector
   $[0,0,...,0,1]^T\in\RR^{n+1}$. Explicitly the matrix form is given by
   \bq 
   x \, \left[  
   \begin{array}c 
   P_0(x) \\ 
   P_1(x) \\ 
   P_2(x) \\ 
 \vdots \\ 
   P_{n-1}(x) \\
   P_n(x)  
   \end{array} 
   \right]  
   =  
    \left[ 
   \begin{array}{ccccccc}  
   \beta_0 & \alpha_0 & 0&0& \ldots &0&0 \\ 
   \gamma_1 & \beta_1 & \alpha_1&0& \ldots &0 &0 \\ 
   0&\gamma_2 & \beta_2 & \alpha_2& \ldots &0&0 \\ 
   \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 
   0&0&0&0& \ldots & \beta_{n-1} & \alpha_{n-1} \\ 
   0&0&0&0& \ldots &\gamma_{n} & \beta_{n}   
   \end{array} 
   \right] 
   \left[  
   \begin{array}c 
    P_0(x) \\ 
   P_1(x) \\ 
   P_2(x) \\ 
 \vdots \\ 
   P_{n-1}(x) \\
   P_n(x) 
  \end{array} 
  \right]  
  + \alpha_n P_{n+1}(x)  \left[  
  \begin{array}c 
   0 \\ 
   0 \\ 
   \vdots \\ 
   1 \\ 
  \end{array} 
  \right].
  \label{ttrr-mf} 
  \eq 
   If we denote by $\{ x_{nj} \}_{1\leq j \leq n}$ the zeros of the polynomial $P_n$,  
   then from (\ref{ttrr-mf}) we conclude that    
    each $x_{nj}$ is an eigenvalue of the corresponding tridiagonal matrix 
    of order $n$ and $[P_0(x_{nj}),..., P_{n-1}(x_{nj})]^T$ is 
    the associated eigenvector. \\ 
 
  When $\alpha_n =1$ and $\gamma_n >0$ for all $n=1,2,\ldots$ the (monic)  
polynomials $\{P_n\}_{n\geq0}$ defined by the recurrence relation 
 (\ref{rrtt})  arise as denominators  
of the approximants of the continued fraction  
$$  
 \displaystyle\frac{1}{x-\beta_0 - \displaystyle 
 \frac{\gamma_1}{x-\beta_1 - \displaystyle\frac{\gamma_2}{x- 
 \beta_2 - \cdots - 
 \displaystyle\frac{\gamma_{n-1}}{x-\beta_{n-1} - 
  \displaystyle\frac{\gamma_n}{x - \beta_n - 
 \ba{c}  \\   \ddots \ea}}}}}\,\,.   
 $$ 
Under these conditions, by Favard's theorem \cite{Chihara}, $\{P_n\}_{n\geq0}$ constitutes an   
orthogonal polynomial sequence  with respect to a positive definite moment  
functional, and if the moment problem associated with the continued fraction is determined, then 
this linear functional can be characterized by some distribution function, i.e.,  
a function $\sigma :\RR \rightarrow \RR $ which is nondecreasing, it has 
infinitely many points of increase and all the moments 
$\int_{-\infty}^{+\infty} x^{2n}d\sigma (x)\, , \, n=0,1,2,\ldots,$   
are finite. The numerators of the  continued fraction, denoted by 
 $\{P^{(1)}_n\}_{n\geq0}$ , can be given by the shifted recurrence relation 
$$ 
   xP^{(1)}_n(x)= P^{(1)}_{n+1}(x)+\beta_{n+1}P^{(1)}_n(x)+ 
\gamma_{n+1}P^{(1)}_{n-1}(x) \quad , \quad n\geq0 \quad , 
$$ 
with initial conditions $P^{(1)}_{-1}=0$ and $P^{(1)}_0=1$.   
This continued fraction converges to a function $F(z)$ and the  
general theory of the moment problem ensures that $F$ is analytic in the  
complex plane with a cut along the support of $\sigma$ (i.e., the set of points of  
increase of $\sigma$). 
This fact can be summarized by Markov-Stieltjes's theorem 
\begin{equation} 
-\lim_{n\rightarrow +\infty}\frac{\mu_0 P^{(1)}_{n-1}(z)}{P_n(z)}= 
F(z)\equiv \int_{-\infty}^{+\infty} \frac{d\sigma (x)}{x-z}   , \quad 
z\in \CC \backslash \mbox{supp$(\sigma)$} \quad , 
\label{Markov} 
\end{equation} 
where $\mu_0=\int_{-\infty}^{+\infty} {d\sigma (x)}$ is the first moment  
of the distribution $\sigma(x)$ and $F$ is  the Stieltjes function  
of the distribution function $\sigma$, defined according to (\ref{stieltjes}). 
Now, the function $\sigma(x)$ 
can be recovered  from (\ref{Markov}) by applying the Stieltjes inversion formula 
$$ 
\sigma(x)-\sigma(y)= 
\lim_{\epsilon \to 0^+} \frac{1}{\pi } \int_{y}^{x} 
Im \ F(t+i\epsilon)\ dt \quad , 
$$ 
where it is assumed that $\sigma$ is normalized in the following way 
$$ 
\sigma(-\infty)=0 \, ,\quad \sigma(x)=\frac{\sigma(x+0)+\sigma(x-0)}{2} \quad . 
$$ 
 
\subsection{The Chebyshev polynomials of the second kind.} 
\esp In this paper we will consider  the monic Chebyshev polynomials of second kind 
$\{ U_n(x) \}_{n\geq0}$. They satisfy the three-term recurrence  
relation  
$$ 
U_{-1}=0 \quad , \quad U_0=1 \quad , \quad  
xU_n(x)=U_{n+1}(x)+\frac{1}{4}U_{n-1}(x)   , \quad n=0,1,2,...\,\, . 
$$ 
They can also be expressed in terms of trigonometric polynomials 
in $\cos \theta$ as 
$$ 
U_n(x)=2^{-n}\frac{\sin (n+1)\theta}{\sin \theta}  , \quad x=\cos \theta .
$$ 
They are orthogonal with respect to the distribution function 
$$ 
d  \sigma_U(x)=\frac{2}{\pi}\sqrt{1-x^2} \ dx \quad , \quad  
\mbox{\rm supp($\sigma_U$)}= [-1,1]   . 
$$ 
The corresponding Stieltjes function is \cite[page 176]{van}
$$ 
F_U(z)=\frac{-2}{z+\sqrt{z^2-1}}=-2(z-\sqrt{z^2-1}) \quad , \quad z\in \CC \backslash[-1,1] 
$$ 
where the square root is such that $| z+\sqrt{z^2-1} | >1$ whenever  
$z\not\in [-1,1]$. Finally, we point out the second order differential equation satisfied by 
$U_n$, 
\bq 
(1-x^2)U^{\prime\prime}_n(x)-3x U^{\prime}_n(x) +n(n+2)U_n(x)=0, \quad  n=0,1,2,\ldots
\label{edso-Un} 
\eq 
\subsection{ The eigenvalues of a tridiagonal 2-Toeplitz matrix.}
%% \subsection{The case of a finite 2-periodic Jacobi matrix.} 
\esp Theorem {\rm \ref{solq1}} can be applied to the determination of the
eigenvalues of a finite tridiagonal 2-Toeplitz matrix $B_{n+1}$, which has the general  form  
   \begin{equation} 
   B_{n+1}= 
   \left[ 
   \begin{array}{cccccc}  
   a_1 & b_1 & 0 & 0 &0& \ldots  \\ 
   c_1 & a_2 & b_2 & 0 &0& \ldots  \\ 
   0 & c_2 & a_1 & b_1 &0& \ldots   \\ 
   0 & 0 & c_1 & a_2 & b_2 & \ldots   \\ 
   0 & 0 & 0 & c_2 & a_1  & \ldots   \\ 
   \vdots & \vdots & \vdots & \vdots & \vdots & \ddots  \\ 
   \end{array} 
   \right] \in \bkC^{(n+1, n+1)}  
   \label{matri}  
   \end{equation} 
   where we assume that $b_1$, $b_2$, $c_1$ and $c_2$ are positive  
   numbers. The eigenvalues and eigenvectors of such a matrix were computed by 
   M.J.C.Gover \cite{Gover} and a different proof, using arguments from the Theory of 
   Orthogonal Polynomials was provided by Marcell\'an and Petronilho \cite{PacoZe5} .  

   \begin{theorem} The eigenvalues of the  
   tridiagonal 2-Toeplitz matrix $B_{2n+1}$ defined by {\rm (\ref{matri})} 
    are $a_1$ and the solutions of the quadratic equations 
   $$ 
   (\lambda -a_1)(\lambda -a_2)-\left[  b_1c_1+b_2c_2+2\sqrt{b_1b_2c_1 c_2}  
   \cos \frac{k\pi}{n+1} \right]=0 \ , \ k=1, \ldots , n \ , 
   $$  
   and the eigenvalues of $B_{2n}$ are the solutions of the quadratic equations  
  $$ 
 (\lambda -a_1)(\lambda -a_2)-\left[  b_1c_1+b_2c_2+2\sqrt{b_1b_2c_1 c_2}  
 \cos \theta_{nk} \, \right]=0 \ , \ k=1, \ldots , n \ , 
 $$  
  where  $\theta_{nk}$'s are the nonzero solutions of the trigonometric  
 equation \\
 $ \sqrt{b_1c_1} \sin [(n+1)\theta] + \sqrt{b_2c_2} \sin (n\theta) =0 \quad  
 (0<\theta<\pi) \ . $  \hfill{\BBox} 
   \label{vp} 
 \end{theorem} 
 
  Since $b_i >0$ and $ c_i > 0$ for $i=1,2$ then there exists an OPS, $\{ S_n \}_{n\geq0}$ , 
 such that $B_n$ is the corresponding tridiagonal  matrix of order $n$. Let $\{ Q_n \}_{n\geq0}$ be 
the corresponding monic OPS. Then  
$$ 
Q_{2n}(x)=(b_1 b_2)^n S_{2n}(x),  \quad\quad Q_{2n+1}(x)=b_1(b_1 b_2)^n S_{2n+1}(x)  . 
$$ 
Moreover, according to the results in \cite{PacoZe5} , $\{ Q_n \}_{n\geq0}$ can be obtained 
 by a quadratic polynomial mapping on a linear transformation of the monic Chebyshev  
polynomials of second kind. In fact, putting 
$$ 
T(x)=(x-a_1)(x-a_2) , \quad \alpha=2\sqrt{b_1b_2c_1c_2}, \quad \beta=b_1c_1+b_2c_2 , 
$$ 
we have $Q_{2n+1}(x)=(x-a_1)P_n(T(x))   , \quad  Q_{2n}(x)=R_n(T(x))$, where 
$\dst P_n(x)=\alpha^n  U_n \left( \frac{x-\beta}{{\alpha}}\right)$, 
$R_{n}(x)=P_n(x)+b_2 c_2 P_{n-1}(x)$, for all $n=0,1,2,\ldots$. Notice that $P_n$ is orthogonal
 with respect to  the distribution function  
$$ 
\sigma_P(x)=\sigma_U \left( \frac{x-\beta}{\alpha}  \right)  ,     \quad 
\mbox{\rm supp($\sigma_P$)}= [ \beta-\alpha ,\beta+\alpha ] \ , \mbox{ so that }
d\sigma_P(x)=  \frac{2}{\pi \alpha^2} 
\sqrt{\alpha^2 -(x-\beta)^2} \ dx \ . 
$$ 
 From \cite{PacoZe5} the polynomials of the first kind of the 
sequence $\{ Q_n \}_{n\geq0}$ are  $Q^{(1)}_{2n-1}(x)=(x-a_2)R^{(1)}_{n-1}(T(x))$, 
$Q^{(1)}_{2n}(x)=P^{(1)}_{n}(T(x))$, the Stieltjes function of $\{ Q_n \}_{n\geq0}$ being 
\begin{equation} 
F_Q(z)=\frac{M}{a_1-z}+\frac{b_1 c_1}{z-a_1}[F_P(T(z))-F_P(0)] \quad , 
\label{ST1} 
\end{equation} 
where $F_P$ denotes the Stieltjes function associated with $\sigma_P$, 
$M=\mu_0-b_1c_1F_P(0)$ and  $\mu_0$ is the first moment of $\sigma_P$. Clearly,  
$\mu_0=\int_{\beta-\alpha}^{\beta+\alpha}d\sigma_P(x)=1$. Furthermore, 
$$ 
F_P(z) = \displaystyle{ \frac{1}{ {\alpha}}  
F_U\left( \frac{z-\beta}{\alpha} \right) }  
= \frac{-2}{\alpha^2}\left( z-\beta - \sqrt{(z-\beta)^2-\alpha^2}  \right) \ , 
$$ 
where the square root is such that  
$|z-\beta + \sqrt{(z-\beta)^2-\alpha^2}|>\alpha$ whenever  
$z\not\in [ \beta-\alpha ,\beta+\alpha ] $. 
Since $0\not\in [ \beta-\alpha,\beta+\alpha ] $ for $b_1c_1\neq b_2c_2$  
(this is no a restriction, because the case $b_1c_1= b_2c_2$ corresponds to  
constant values along the diagonal of the corresponding  tridiagonal  matrix),  
elementary computations give 
$F_P(0)= { \min \{ b_1 c_1 , b_2 c_2\}}/{ b_1 c_1 b_2 c_2 }$, hence 
$M\dst =1 - \frac{\min \{ b_1 c_1 , b_2 c_2 \}}{  b_2 c_2  }$.  
It turns out from (\ref{ST1}) that the Stieltjes function of  
$\{ Q_n \}_{n\geq0}$ reads as 
\bq
F_Q(z)=\frac{M}{a_1-z}- 
\frac{1/b_2 c_2\alpha}{z-a_1}\left[\frac{1}{2} 
\left(T(z)-\beta-\sqrt{(T(z)-\beta)^2-\alpha^2}  \right) 
+\min \{ b_1 c_1 , b_2 c_2 \} \right] \ . 
\label{ex-Stj}
\eq 
From this we find (see \cite{PacoZe5}) that $\{ Q_n \}_{n\geq0}$ is  
orthogonal with respect to the distribution function   
$$ 
\begin{array}{lll} 
d  \sigma_Q(x) & = & \displaystyle{ M \delta(x-a_1) + \frac{1}{2\pi \, b_2 c_2} \frac{1}{|x-a_1|}  
 \sqrt{4b_1 b_2 c_1 c_2 -(T(x)-b_1c_1-b_2c_2)^2} \ dx  } 
 \quad , 
\end{array} 
$$ 
the support being the union of two intervals if $M=0$ and the union of two intervals with 
a singular point if $M>0$ : either 
$$ 
\mbox{\rm supp($\sigma_Q$)} = \left\{ \ba{ll} 
 T^{-1}( \mbox{\rm supp($\sigma_P$)} )  & 
    \mbox{\rm if } \quad b_1 c_1 \leq b_2 c_2  \\
     & \\  
 T^{-1}( \mbox{\rm supp($\sigma_P$)} )  \cup \{ a_1 \} 
&  \mbox{\rm if } \quad b_1 c_1 > b_2 c_2  
\ea \right. \esp .
$$ 
We notice that 
$$ 
\begin{array}{l}
T^{-1}( \mbox{\rm supp($\sigma_P$)} ) = T^{-1}(  [ \beta-\alpha ,\beta+\alpha ]  )=
\dst[\frac{a_1+a_2}{2}-s,\frac{a_1+a_2}{2}-r] \cup  
[\frac{a_1+a_2}{2}+r,\frac{a_1+a_2}{2}+s] \ , 
\end{array}
$$ 
where $\dst  r=\left( \left| \sqrt{b_1c_1}-\sqrt{b_2c_2} \right|^2 + 
 \left| \frac{a_1-a_2}{2} \right|^2  \right)^{1/2} , \,\,\,
 s=\left( \left| \sqrt{b_1c_1}+\sqrt{b_2c_2} \right|^2 + 
 \left| \frac{a_1-a_2}{2} \right|^2  \right)^{1/2}$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{ The eigenvalues of a tridiagonal  3-Toeplitz matrix.}
\esp Theorem {\rm \ref{solc3}} can be applied 
to the determination of the eigenvalues of a finite tridiagonal 3-Toeplitz 
matrix, which has the general
form 
$$
B_{n+1}=\left[ 
\begin{array}{llllllll}
a_1 & b_1 & 0 & 0 & 0 & 0 & 0 & \cdots \\ 
c_1 & a_2 & b_2 & 0 & 0 & 0 & 0 & \cdots \\ 
0 & c_2 & a_3 & b_3 & 0 & 0 & 0 & \cdots \\ 
0 & 0 & c_3 & a_1 & b_1 & 0 & 0 & \cdots \\ 
0 & 0 & 0 & c_1 & a_2 & b_2 & 0 & \cdots \\ 
0 & 0 & 0 & 0 & c_2 & a_3 & b_3 & \cdots \\ 
0 & 0 & 0 & 0 & 0 & c_3 & a_1 & \cdots \\ 
\vdots & \vdots & \vdots & \vdots & \vdots & 
\vdots & \vdots & \ddots
\end{array}
\right]   \in \bkC^{(n+1, n+1)}  
$$
Again, we assume that $B_{n+1}$ is irreducible, i.e., 
$b_ic_i\neq 0$ ($i=1,2,3$). 
Given $B_{n+1}$, define a sequence of monic polynomials 
$\{P_n\}_{n\geq 0 }$ by 
\[
P_n(x)=\left( 2\sqrt{b_1b_2b_3c_1c_2c_3}\right)^n
\widehat{U}_n\left( 
\frac{x-b_1c_1-b_2c_2-b_3c_3}
{2\sqrt{b_1b_2b_3c_1c_2c_3}}\right) 
\, ,\quad
n\geq 0 \, .
\]
Now, define
a polynomial $\pi_3$ as
$$
\begin{array}{l}
\pi _3(x)=(x-a_1)(x-a_2)(x-a_3)-
(b_1c_1+b_2c_2+b_3c_3)(x-a_3)+ \\ 
\qquad \qquad +b_1c_1+b_2c_2+
b_3c_3-b_2c_2(a_3-a_1)-b_3c_3(a_3-a_2)\, {,}
\end{array}
$$
and denote by $a$ and $b$ the roots of the 
quadratic equation $(x-a_1)(x-a_2)-b_1c_1=0$. 
Thus, using Theorem \ref{solc3}, it can be 
shown  that the eigenvalues 
of $B_n$ are the zeros of a  MOPS 
$\{Q_n\}_{n\geq 0 }$ such that 
$$
\begin{array}{l}
Q_{3n}(x)=P_n(\pi _3(x))+b_3c_3(x-a_2)
P_{n-1}(\pi _3(x)) \\ 
Q_{3n+1}(x)=(x-a_1)P_n(\pi _3(x))+
b_1c_1b_3c_3P_{n-1}(\pi _3(x)) \\ 
Q_{3n+2}(x)=(x-a)(x-b)P_n(\pi _3(x))\, 
\end{array}
$$
(the parameters for the corresponding 
three-term recurrence relation
are $\tilde \beta _{3n}=a_1$, 
$\tilde \beta _{3n+1}=a_2$,
$\beta _{3n+2}=a_3$, 
$\tilde \gamma _{3n+1}=b_1c_1$, 
$ \tilde \gamma _{3n+2}=b_2c_2$ and
$\tilde \gamma _{3n+3}=b_3c_3$).
 In particular, the eigenvalues of 
 $B_{3n+2}$ are $a$ , $b$ and the 
roots of the cubic
equations 
\[
\pi _3(x)=b_1c_1+b_2c_2+b_3c_3+
2\sqrt{b_1b_2b_3c_1c_2c_3}\cos \frac{k\pi }{n+1}
\quad ,\quad k=1,\ldots ,n \, .
\]


\section{The Chain Model.} 
\esp    The Chain Model has been applied in several  areas of  
 Modern Physics: Solid State Physics \cite{hay-pap}, Nuclear Physics 
  \cite{wwcm},  Quantum Mechanics \cite{haydock-nevai}, etc. 
  In this section we will describe the General Chain Model. For more details  
  see \cite{haydock}-\cite{haydock-nevai} and references contained  
 therein. 
\subsection{Physical Motivation.}
 \esp It is well known the importance of the eigenvalue problem  
 for symmetric matrices since many quantum systems can be described  
 using such kind of matrices (see \cite[Chapter 2, Section 6]{von-n}, and  
 also \cite{messiah}). \\ 

 In general, the computation of the eigenvalues (and also the eigenvectors) 
 for an arbitrary matrix is very difficult  since it is  
 equivalent to the problem of finding the  roots of a polynomial 
 of degree $n$ and it is  not posible to obtain, for an arbitrary polynomial,  
 an analytic expressions. Moreover, the numerical algorithms (Newton's Method, etc) 
 are, in general, unstable for large $n$. For this reason, in several   
  areas of Modern Physics simpler models have been introduced,  
 the so called {\em Chain Models} (see  \cite{hay-pap},  
  \cite{wwcm}, \cite{haydock} and references contained  
 therein).   The Chain Model is, in general, 
 easier to solve and numerically it is more stable and economic. The reason  
 is that such a model is connected with   Jacobi (tridiagonal symmetric)  
 matrices  which have a lot of {\em good} properties \cite{jac-mat}.\\  

 The physical interest of the Chain model consists in the fact that  
 any quantum model can be transformed into the  
 corresponding  {\em Chain Model}. The main physical idea of the model is  
  that in order to find the local density of a given state it is sufficient to  
 consider the neighboring states. This means that the local  
 density depends strongly on the  neighboring states  while the more  
 distant ones have smaller effect. The solution of this  problem is the  
 determination  
 of linear  combinations of the local states such that the system pass  
 through each state as it diffuses away from the local initial state. 
 The local density for a basis state $|\Phi_0\rangle$ is determined by the sequence  
 of {\em orthogonal states}(orbits) $|\Psi_n\rangle$ where  
 $|\Psi_0>\equiv|\Phi_0>$.   
 The orthogonal states (orbits) $|\Psi_n>$ must satisfy the time 
 independent Schr\"odinger equation 
 \bq 
 \HH |\Psi_n> = b_{n+1}|\Psi_{n+1}> +  a_{n}|\Psi_{n}>+  b_{n}|\Psi_{n-1}>, 
 \label{H-CM} 
 \eq 
 where $\HH$ is the Hamiltonian of the model. Here the parameters 
 $a_n,\,\,b_n$ describe the coupling of each orthogonal state with  
 itself and its neighbors. In \cite{haydock} has been considered the 
 {\em constant chain model} corresponding to the special cases when  
 the sequences $\{a_n\}$ and $\{b_n\}$ are constants equal to  
 $a$ and $b$, respectively. Here  we will consider a generalization  
 of the constant model: The 2-Periodic Chain Model corresponding to it  
 where  the sequences  $\{a_n\}$ and $\{b_n\}$  are periodic  
 sequences with period 2, i.e.,  $\{a_n\}=\{a,b,a,b,\cdots\}$ and  
 $\{b_n\}=\{c,d,c,d,\cdots\}$. For such a model it is possible   
 to obtain analytic formulas for  the values of the energy (eigenvalues) 
 of $\HH$ and the corresponding wave functions. 
 
 \subsection{Notation.} 
 \esp Before define the Chain Model let us introduce some notation.  
 Assume that the set of wave functions $\{|\Phi_k>\}_{k\geq 0}$ is a basis of our space 
 of states  not necessarily  orthogonal in the sense that  the inner product 
$<\Phi_m|\Phi_n>$ $\equiv$ $ \dst\int \overline{\Phi_m}\Phi_n\,d\tau$ could be different from 
 zero for $m\neq n$.  In the following, we will suppose that any state $|u>$  
 of the system can be represented as a linear combination of the  
 basis $\{|\Phi_k>\}_{k\geq 0}$, i.e., $ |u >=\sum_{k=0}^\infty u_k |\Phi_k> 
 $  and we will represent it by the infinite column vector ${\bf u}$ with  
 coordinates $u_k$. The  
 action of the Hamiltonian $\HH$ (\ref{H-CM}) on a basis state  
 $|\Phi_n>$ is represented, in general, by the Hamiltonian  matrix  
 $\hh$ defined by means of the parameters  $\hh_{mn}$ in such a way that   
 \bq 
 \HH |\Phi_n>=\sum_{m=0}^\infty\hh_{mn}|\Phi_m> . 
 \label{me-H} 
 \eq 
 The $\hh_{mn}$  are, in general, different from the matrix elements of 
 the Hamiltonian $\HH_{mn}$ defined by $\HH_{mn}= <\Phi_m| \HH |\Phi_n>$.  
 The matrix elements $\HH_{mn}$ of the Hamiltonian constitute a 
 self-adjoint matrix (Hermitian), i.e.,  $ \overline{\HH_{mn}^T} = \HH_{mn}$. Here, by  
 $A^T$ we denote the transpose of a matrix $A$ and by $\overline{A}$ the  
 complex conjugate of A. We also need the overlap matrix $\SS$ defined by the 
 matrix elements $ \SS_{mn}=<\Phi_m|\Phi_n>$,  which is also a  self-adjoint matrix. 
 The elements $\HH_{mn}$ and $\hh_{mn}$ are related to each other by 
 the expression 
 \bq 
 \HH_{mn}= [\SS\hh]_{mn} 
 \label{rela-h-H} 
 \eq  
In some cases, it is known  $\SS$ and $[\HH_{mn}]$. Then,  
 for $\HH$ the following  formula holds $ \hh= \SS^{-1} [\HH_{mn}]$. 
 Using the above notation the time independent  
 Schr\"odinger equation can be written in the matrix form 
 \bq 
 \hh {\bf u}_\alpha= \ener_\alpha {\bf u}_\alpha, 
 \label{s-eq-mf} 
 \eq 
 where $\alpha$ is an index containing all quantum numbers. 
 
\subsection{The Chain Model.} 
 \begin{dff} 
 The Chain Model is a quantum model determined by a sequence of orthonormal  
 orbits (states) $\{{\bf u}_0, {\bf u}_1, \cdots  \}$ and two sets of real  
 parameters $\{a_1,a_2,\cdots\}$ and 
 $\{b_1,b_2,\cdots\}$, which describe the action of the Hamiltonian  
 $\hh$ on the orbitals by a symmetric three-term recurrence relation 
 of the form 
 \bq 
 \hh {\bf u}_n = b_{n+1} {\bf u}_{n+1} + a_n {\bf u}_n + b_n {\bf u}_{n-1}. 
 \label{def-CM}  
 \eq 
 The sequence  $\{{\bf u}_0, {\bf u}_1, \cdots  \}$ may be finite or 
 infinite. In the first case we need to take the orbitals  
 ${\bf u}_{-1}$ and  ${\bf u}_{N+1}$ equal to zero. 
 The overlap of the states $\{{\bf u}_n\}$ is defined in terms of the 
 matrix  $\SS$ by formula ${\bf u}_n^{+} \SS {\bf u}_m= \delta_{nm}$, where  
  $a^+\equiv \overline{a}^T$ and $\delta_{km}=1$ for $k=m$, elsewhere $\delta_{km}=0$. 
  This model   is equivalent to expressing the matrix of $\hh$ as a Jacobi (tridiagonal 
 symmetric) matrix \cite{haydock} 
  \begin{equation} 
 \hh=   \left( 
   \begin{array}{cccccc}  
   a_0   & b_1    & 0      & 0      &  \ldots \\ 
   b_{1} & a_{1}  & b_{2}  & 0      &  \ldots \\ 
    0    & b_{2}  & a_{2}  & b_3    &  \ldots \\ 
    0    &   0    & b_{3}  & a_3    &  \ldots \\ 
  \vdots & \vdots & \vdots & \vdots & \ddots   
   \end{array}\right) . 
   \label{matrix-CM}  
   \end{equation} 
 \end{dff} 

 The  chain model can be interpreted as follows: Given a system in the  
 initial state ${\bf u}_0$ it can go from ${\bf u}_0$ to a new state  
 ${\bf u}_1$ which can represent a linear combination of the physical  
 (real) states neighboring ${\bf u}_0$. Similarly, having reached ${\bf u}_1$, 
 the system can go back to the previous state ${\bf u}_0$ or go on  
 to another state ${\bf u}_2$, etc. In other words, the behaviour of the  
 system  is most influenced by the nearer preceding or succeeding states   
  and less influenced by the more distant ones. This  is the mathematical  
 expression of the physical concept of local environment \cite{haydock}. 
 In the following, we will suppose that the states are ordered  
 by the sequence  $\{ {\bf u}_0, {\bf u}_1, {\bf u}_2,\cdots\}$ where  ${\bf u}_0$ is the  
 initial state   and  the others  ${\bf u}_k,\,\, k\geq 1$ are the succeeding states. 
  We will say that a state  ${\bf u}_k$ is even or odd if $k=2n$ or $k=2n+1$, 
  $n=0,1,2,\cdots$, respectively.   

 \subsection{The 2-Periodic Chain Model.} 
 \esp We will suppose that the sequences of {\em coupling constants}  
 $\{a_n\}$ and $\{b_n\}$  are periodic sequences with period 2, i.e.,  $\{a_n\}=\{a,b,a,b,\cdots\}$ and  
 $\{b_n\}=\{c,d,c,d,\cdots\}$. Then the matrix (\ref{matrix-CM}) becomes 
  \begin{equation} 
 \hh=  \left( 
   \begin{array}{cccccc}  
  a & c & 0  & 0 & 0 & \ldots   \\ 
  c & b & d  & 0 & 0 & \ldots   \\ 
  0 & d & a  & c & 0 & \ldots   \\ 
  \vdots & \vdots & \vdots &  \vdots & \vdots & \ddots  \\ 
 \end{array} 
   \right). 
   \label{mat-modelo}  
   \end{equation} 
 These coupling constants are defined in such a way that the interaction between 
  different  states leads us to a Hamiltonian  $\HH $ with the following properties  
 ($|u_k>\equiv {\bf u}_k$) 
 \bq 
 \ba{l} 
 h_{mm}=< u_m|\hh | u_m> = \left\{ \ba{ll} a & \mbox{if $m$ is even} \\ 
                                      & \\ 
                                      b & \mbox{if $m$ is odd} \ea\right. \\ \\ 
  h_{mn}= < u_m|\hh| u_n > =  
  \left\{ \ba{ll} c &  \mbox{for $m=2n-1$, $k=2n$ } \\ 
       & \\ 
     d & \mbox{ for $m=2n$, $k=2n+1$ } \ea \right. \ea 
 \label{def-H} 
 \eq 
 Notice that the order of the interaction and the parity is crucial, i.e., 
 even and odd states have a different behavior. This can be interpreted as  
 follows: the coupling between even or odd states is equal to   
 $a$ and $b$, respectively and not necessarily equal. The coupling between an 
 even state and its preceding or its succeeding is equal to $c$ or $d$,  
 respectively  and not necessarily equal.
 Analogously, for the coupling between odd states and their neighbors,  
 we have coupling $d$ or $c$, respectively, when it is coupled with the 
 preceding or the succeeding state. The scheme represented in 
 Figure \ref{esq}  can be useful to understand this 2-Periodic Chain Model. 


{  \begin{figure} 
\vspace*{-.5cm} 
%%%%%%%\begin{center} 
\begin{picture}(340,150)(0,0) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% linea de estados 
\put(-10,120){\mbox{ $\dots$ }} 
\put(20,120){\line(1,0){323}} 
\put(355,120){\mbox{ $\dots$ }} 
\put(20,120){\circle*{5}} 
\put(40,120){\circle*{5}} 
\put(80,120){\circle*{5}} 
\put(100,120){\circle*{5}} 
\put(130,100){\line(0,1){40}} 
\put(130,100){\line(1,0){100}} 
\put(140,120){\circle*{5}} 
\put(160,120){\circle*{5}} 
\put(200,120){\circle*{5}} 
\put(220,120){\circle*{5}} 
\put(230,100){\line(0,1){40}} 
\put(130,140){\line(1,0){100}} 
\put(260,120){\circle*{5}} 
\put(280,120){\circle*{5}} 
\put(320,120){\circle*{5}} 
\put(340,120){\circle*{5}} 
\put(180,85){\mbox{\Huge$\Downarrow$}} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% detalle 
\put(70,25){\oval(20,50)} 
\put(130,25){\oval(20,50)} 
\put(65,55){\mbox{a}} 
\put(125,55){\mbox{b}} 
\put(95,5){\mbox{c}} 
\put(185,5){\mbox{d}}  
\put(250,25){\oval(20,50)} 
\put(310,25){\oval(20,50)} 
\put(275,5){\mbox{c}} 
\put(245,55){\mbox{a}} 
\put(305,55){\mbox{b}} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\put(75,0){\line(1,0){235}} 
 \put(75,0){\vector(1,0){50}} 
\put(125,0){\vector(-1,0){50}} 
\put(135,0){\vector(1,0){110}} 
\put(245,0){\vector(-1,0){110}} 
\put(255,0){\vector(1,0){50}} 
\put(305,0){\vector(-1,0){50}} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\put(55,-10){\mbox{\footnotesize$2m-1$}} 
\put(70,0){\circle*{5}} 
\put(123,-10){\mbox{\footnotesize$2m$}} 
\put(130,0){\circle*{5}} 
\put(235,-10){\mbox{\footnotesize$2m+1$}} 
\put(250,0){\circle*{5}} 
\put(295,-10){\mbox{\footnotesize$2m+2$}} 
\put(310,0){\circle*{5}} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\put(10,20){\mbox{$\dots$}} 
\put(10,50){\mbox{$\dots$}} 
\put(10,0){\mbox{$\dots$}} 
\put(350,50){\mbox{ $\dots$ }} 
\put(350,20){\mbox{ $\dots$ }} 
\put(350,0){\mbox{ $\dots$ }} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%% cuadro 
\put(30,80){\line(0,-1){100}} 
\put(30,80){\line(1,0){310}} 
 
\put(340,-20){\line(0,1){100}} 
\put(30,-20){\line(1,0){310}} 
 
 
\end{picture} 
%%%%%%\end{center} 
  \vspace*{1cm} 
 \caption{Scheme of the 2-Periodic Chain Model.} 
\vspace{0.2cm} 
\label{esq} 
\end{figure}} 
 \subsection{The Eigenvalues and Eigenstates of the 2-Periodic Chain.} 
 \esp In order to obtain the  eigenvalues and eigenstates of the 2-Periodic Chain we will  
 suppose that  the solution ${\bf u}$ of the  Schr\"odinger equation (\ref{s-eq-mf})  
 can be written as a linear combination of the states  
  $\{ {\bf u}_0, {\bf u}_1, {\bf u}_2,\cdots\}$, i.e.,  
 \bq 
  {\bf u}=  \sum_{k=0}^\infty C_k {\bf u}_k . 
 \label{l-c-ev} 
 \eq 
 Then,  (\ref{s-eq-mf}) can be rewritten in the form 
  \begin{equation} 
   \left(  \begin{array}{cccccc}  
  a & c & 0  & 0 & 0 & \ldots   \\ 
  c & b & d  & 0 & 0 & \ldots   \\ 
  0 & d & a  & c & 0 & \ldots   \\ 
  \vdots & \vdots & \vdots &  \vdots & \vdots & \ddots  \\ 
 \end{array} 
 \right) \left( \ba{c} C_{1}\\ C_{2}\\ C_{3}\\ \vdots  \ea \right)= 
  \ener 
  \left(  \ba{c} C_{1}\\ C_{2}\\ C_{3}\\ \vdots \ea \right), 
   \label{sis-mat-esp}  
   \end{equation} 
 where $C_{k}$ are the coefficients in the linear combination (\ref{l-c-ev}), and $\ener$ 
 is the corresponding eigenvalue of the matrix Hamiltonian $\hh$. Here, it is important
  to remark that we need to consider finite or infinite chains. The solution of this  
 eigenvalue problem for the finite chain is given by theorem \ref{vp}, which leads us to the 
 following two cases:

\subsubsection {Case of $N=2n+1$ states in the Chain.} 
 In this case, we have $a_1=a,\, a_2=b, \, b_1=c_1=c$ and $b_2=c_2=d$ and then the  
 eigenvalues $\ener$ of (\ref{sis-mat-esp}) are the following 
 \bq 
 \ba{l} 
 \ener_0=a , \esp 
 \ener_{\pm k}= \displaystyle \frac{a+b}{2} 
 \pm \sqrt{ \frac{(a-b)^2}{4} + c^2+ d^2 + 2cd \cos{ 
  \mbox{$ \frac{k\pi}{n+1} $ } } },\esp k=1,2,...,n 
 \ea 
 \label{caso2n+1} 
 \eq 
The particular case $a=b$ and $c= d$ gives us $ \ener_0=a$, 
$ \ener_{\pm k}= \displaystyle a \pm 2c   \cos{\mbox{$\frac{k\pi}{2(n+1)}$ }}$ para 
$k=1,2,...,n $.
   

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 \subsubsection {Case of  $N=2n$ states in the Chain.} 
 In this case, we have $a_1=a,\, a_2=b, \, b_1=c_1=c$ and $b_2=c_2=d$ and then the 
  eigenvalues  
 $\ener$ of (\ref{sis-mat-esp}) are the following 
 \bq 
 \ba{l} 
 \ener_{\pm k}= \displaystyle \frac{a+b}{2} 
 \pm \sqrt{ \frac{(a-b)^2}{4} + c^2+ d^2 + 2cd \cos{ \theta_{Nk} } },\esp k=1,2,...,n 
 \ea 
 \label{caso2n} 
 \eq 
  where   $\theta_{Nk}$'s are the nonzero solutions of the trigonometric  
 equation 
 $$ 
 c \sin [(n+1)\theta] + d \sin (n\theta) =0 \quad . 
 $$ 
The particular case $a=b$ and $c= d$ gives us the values
$\ener_{\pm k}= \displaystyle a \pm 2c \cos{\mbox{$ \frac{k\pi}{2n+1}$} }$.  

\subsection{The Green Function $G_0(\ener)$.} 
 \esp For this model it is possible to obtain analytic formulas for the  
general diagonal Green function $G_{0}(\ener)\equiv G_{00}(\ener)$. 
 The Green function (usually called the {\em projected density of states}) 
$G_{nm}(\ener)$ between two states ${\bf u}_n$ and ${\bf u}_m$ is  
 formally defined by $G_{nm}(\ener)={\bf u}_n^+ \SS(\ener-\hh)^{-1} {\bf u}_m$.
The quantity $G_{nn}(\ener)$  is the density of states  
in energy with each state weighted by its projection 
on the atomic orbital ${\bf u}_n$.  
In \cite{haydock}, \cite{hay-pap} it is shown that 
the Green function of the initial state ${\bf u}_0$  
$G_0(\ener)$  for the Chain Model (\ref{def-CM}) is related to the 
 continued fraction  
\bq  
 G_0(\ener)= \displaystyle\frac{1}{\ener-a_0 - \displaystyle 
 \frac{b_1^2}{\ener-a_1 - \displaystyle\frac{b_2^2}{\ener- 
 a_2 - \ba{c}  \\ \ddots \ea }} } \,,  \esp \ener\in \CC. 
\label{def-G0} 
\eq 
 For the finite Chain Model, this continued fraction reduces to 
 the ratio of two polynomials which conforms the well known  
 Pad\'e Approximant of order $n$ of the infinite continued fraction. 
 This and some other results concerning the calculation of the  
 Green function will be considered in detail in the next section.  
 Here we want to remark that the real part of $G_0(\ener)$ describes  
 the response of the system to be driven at a given energy, i.e., 
 the resistence to excitation different from the eigenenergies, although 
the $n_0(\ener)=-\dst\lim_{\epsilon\to 0}\frac{1}{\pi} 
Im \left(G_0(\ener+i\epsilon)\right)$ is the local  
density of the initial state. In \cite{haydock-nevai}, it has been shown the  
 relation of this quantity $n_0(\ener)$ and the electron density in multielectron 
 systems and also in Quantum Mechanics. 
   
 \subsection{The infinite chain.} 
 \esp In the case of the infinite chain, it is possible to obtain an analytic expression 
 for the Green function $G_0(\ener)$ (\ref{def-G0}). In this case, using Rational 
  Approximation Theory \cite{wall}, we obtain that the continued fraction  (\ref{def-G0}) 
  converges to the Stieltjes function associated with the measure of orthogonality of the 
  polynomial sequence $\{Q_n\}_{n\geq 0}$, i.e., $G_0(\ener)= F_Q(\ener)$.  
  In this case, we can obtain the local density $n_0(\ener)$ which coincides 
  with the corresponding measure of orthogonality $d\sigma_Q(\ener)$ 
  (see formula (\ref{ex-Stj})). 
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The semiclassical approximation to the density of the  
distribution of the eigenvalues and their moments.} 
\esp Here, we will apply the   semiclassical or WKB  
approximation \cite{zarzo1}, \cite{zarzo2}, \cite{zarzo3} 
to obtain the density of the distribution  $\rho_n(\ener)$ 
of the eigenvalues of the 2-periodic Hamiltonian (\ref{sis-mat-esp})  
and use the method  developed by  Dehesa  \cite{dehesa1} 
to obtain their moments $\mu_r$ around the origin, i.e., we will calculate the 
quantities  
\bq 
\rho_n(\ener)=\frac{1}{n}\sum_{i=1}^n \delta(\ener-\ener_{n,i}),\esp 
 \mu_r= \displaystyle \frac{1}{n} y_r = \frac{1}{n} \sum_{i=1}^{n} \ener_{n,i}^{r} \quad ,
\label{def-densidad} 
\eq 
where $\ener_{n,i}$, $i=1,2,...,n$,  are the eigenvalues of the  
matrix (\ref{mat-modelo}).  To obtain the WKB-density (and its moments), we will use the fact  
that the eigenvalues of the matrix $B_m$ (\ref{mat-modelo}) coincide with the zeros 
of the  polynomial $Q_m$ associated with the (infinite) Jacobi matrix  
polynomial sequence $\{ Q_k \}_{k\geq 0}$. For odd matrices, i.e., when $B_{2n+1}$, 
the corresponding polynomials are quadratic mappings of the Chebyshev ones. In the  
even case we will have the co-recursive polynomials $R_n$. In both  
cases, it is possible to find the second order differential equation that those 
polynomials satisfy, and then, we can use the semiclassical or WKB approximation 
which allows us to obtain an analytic expression for the density $\rho_n(\ener)$.\\ 
 
   First of all we  point out that, since we are working with a Jacobi matrix, 
  all its eigenvalues  are real, simple and located in the interior of the convex hull of the  
  interval of orthogonality of the associated orthogonal polynomials. 
   This is a necessary condition in order to apply both algorithms.\\ 
    
     Now, let us analyze the so-called semiclassical or WKB approximation  
   (see \cite{zarzo1},\cite{zarzo2} and references contained therein).  Denoting the zeros of  
    a polynomial $\tilde{P}_n(x)$ by $\{x_{n,k}\}_{k=1}^n$, we can define its distribution function 
    $\rho_n(x)$ and their  moments around the origin $\mu_r$ as  
    \bq 
    \rho_n(x)= \frac{1}{n} \sum_{k=1}^n \delta (x-x_{n,k}),\esp 
      \mu_r= \displaystyle \frac{1}{n} y_r = \frac{1}{n} \sum_{i=1}^{n} x_{n,i}^{r}. 
    \label{dist} 
    \eq 
     
     We will follow the  method presented in \cite{zarzo2} in order to obtain 
     the  WKB density of zeros, which is an approximate  expression for the 
     density of zeros of solutions  of any  second order linear 
     differential equation with polynomial coefficients  (SODE) 
     \bq 
     p(x) y''(x) + q(x) y'(x) + r(x) y(x) =0 .
     \label{eqdepar} 
     \eq 
        
     The main result is established in the following   
     \begin{theorem}\label{teoWKB} \cite{zarzo1}, \cite{zarzo2}
     Let $S(x)$ and $\epsilon(x)$ be the functions 
     \bq 
     S(x)= \displaystyle \frac{1}{4 p(x)^2} \{ 2 p(x) (2 r(x) -q'(x)) + 
      q(x)(2 p'(x) - q(x)) \}, 
      \label{espag} 
   \eq 
     \bq 
     \epsilon(x) =\displaystyle \frac{1}{4 [S(x)]^2} 
     \left\{ \frac{5 [S'(x)]^2}{4 [S(x)]} - S''(x)\right\}=\frac{\tilde{P}(x,n)}{\tilde{Q}(x,n)}, 
 \label{eps} 
 \eq 
where $\tilde{P}(x,n)$ and $\tilde{Q}(x,n)$ are polynomials in $x$ as well as in $n$.  
     If the condition $\epsilon(x)<<1$ holds, then, the semiclassical or  
     WKB density of 
     zeros of the solutions of (\ref{eqdepar}) is given by  
     \bq 
     \rho_{WKB}(x)= \frac{1}{\pi} \sqrt{S(x)}, \esp x\in I \subseteq \RR, 
     \label{wkbrho} 
     \eq 
     in every interval $I$ where the function $S(x)$ is positive. \hfill $\BBox$
     \end{theorem} 

   In order to  study the moments of the distribution of eigenvalues of  
   our 2-periodic matrix we  will use a  general method presented in  
   \cite{dehesa1}  for the moments of low order $\mu_r$ of the distribution of zeros 
   (\ref{dist}). Dehesa \cite{dehesa1} has obtained a general formula to find   these quantities.
    We will  apply these  formulas to obtain the general expression for the moments $\mu_1$ 
  $\mu_2$ and $\mu_3$. If we have a tridiagonal symmetric matrix $J_{n}$ where $\alpha_{n}\equiv\gamma_{n+1}$,
   then the first three moments of its eigenvalue distribution are given by the formulas
 \bq
 \ba{l}\dst
 \mu_1=\frac{1}{n}\sum_{k=0}^{n-1} \beta_k, \esp \dst \mu_2= \frac{1}{n} \sum_{k=0}^{n-1} \beta_k^2 + 2 \sum_{k=0}^{n-2}\alpha_k^2, \esp
 \dst \mu_3= \frac{1}{n} \sum_{k=0}^{n-1} \beta_k^3 + 3 \sum_{k=0}^{n-2}\alpha_k^2(\beta_k+\beta_{k+1}). \\
 \ea
 \label{for-mom}
 \eq
Let us to apply the Theorem above to calculate the distribution of eigenvalues of our 
symmetric  Hamiltonian matrix (\ref{sis-mat-esp}).   
Here we wish to point out  that in the odd case the corresponding polynomials 
are  $Q_{2n+1}(\ener)=(\ener-a_1)P_n(T(\ener)) $, so we can consider only the  
distribution of zeros of $P_n$, since for any $n=1,2,...$, $\ener=a$ is always a  
zero of the polynomial and thus an eigenenergy. Furthermore, $P_n(T(\ener))$ is a 
quadratic modification of the Chebyshev polynomials $U_n(\ener)$    
and then  the SODE follows from the classical ones (\ref{edso-Un}) simply making the 
 change of variables $\ener \leftrightarrow T(\ener)$. In fact we deduce that the polynomial
$P_{n}(T(\ener))$ satisfies a SODE (\ref{eqdepar}) with the coefficients 
 
\bq\ba{rl} 
p(\ener)=&  
  \left( 4\,{c^2}\,{d^2} - {{\left( -\left( a\,b \right)  + {c^2} +  
         {d^2} + a\,\ener + b\,\ener - {\,\ener^2} \right) }^2} \right) ,  \\ 
q(\ener)= &3\,{{\left( -a - b + 2\,\ener \right) }}\, 
   \left( -\left( a\,b \right)  + {c^2} + {d^2} + a\,\ener + b\,\ener -  
      {\,\ener^2} \right)  - \\ 
   &   - 2\,\left( 4\,{c^2}\,{d^2} -  
     {{\left( -\left( a\,b \right)  + {c^2} + {d^2} + a\,\ener + b\,\ener -  
          {\,\ener^2} \right) }^2} \right), \\ 
r(\ener)=& n\,\left( 2 + n \right) \,{{\left( -a - b + 2\,\ener \right) }^2} .
\ea 
\label{dif-eq-odd} 
\eq 
For the even case the situation is more complicated since we need to calculate the  
SODE for the co-recursive polynomials $R_n(T(\ener))$. Using the symbolic program {\em Mathematica} 
\cite{wol}, as well as the package {\em PowerSeries} developed by Koepf \cite{koepf}, 
we obtain for the co-recursive polynomials $R_n(\ener)$ a SODE (\ref{eqdepar}) with coefficients 
 \bq\ba{l} 
p(\ener)=\left( 1 + n + {c^4}\,n + {c^2}\,\ener +  
    2\,{c^2}\,n\,\ener \right) \, 
  \left( -1 + {\ener^2} \right) ,  \\  
q(\ener)=  {c^2} + 2\,{c^2}\,n + 3\,\ener + 3\,n\,\ener +  
  3\,{c^4}\,n\,\ener + 2\,{c^2}\,{\ener^2} +  
  4\,{c^2}\,n\,{\ener^2}, \\ 
r(\ener)=  -2\,n + {c^4}\,n - 3\,{n^2} - {n^3} -  
  {c^4}\,{n^3} - {c^2}\,n\,\ener -  
  3\,{c^2}\,{n^2}\,\ener - 2\,{c^2}\,{n^3}\,\ener .
\ea 
\label{dif-eq-corecursivos} 
\eq 
Making the change of variables $\ener \leftrightarrow T(\ener)$ 
($T(\ener)=(\ener-a)(\ener-b)$) in the previous SODE, we find the following  SODE  
for the polynomials $Q_{2n}(\ener)$ 
$$ 
\tilde{p}(\ener) Q''_{2n}(\ener) + \tilde{q}(\ener) Q'_{2n}(\ener) +
\tilde{r}(\ener) Q_{2n}(\ener)=0, 
$$ 
where the  coefficients are given by 
 \bq\ba{l} 
\tilde{p}(\ener)  = 4 c^2 d^2 T''(\ener) p\left(\frac{T(\ener )-a^2-b^2}{2 c d} \right) ,\esp 
\tilde{r}(\ener) = T'(\ener)^3 r \left(\frac{T(\ener)-a^2-b^2}{2 c d} \right),\\ 
\tilde{q}(\ener)   = 2 c d T'(\ener)^2 q \left(\frac{T(\ener)-a^2-b^2}{2 c d} \right) - 
8 c^2 d^2 p \left(\frac{T(\ener)-a^2-b^2}{2 c d} \right). 
\ea 
\label{dif-eq-even} 
\eq 

\begin{figure}[t]
 \vspace*{9cm} 
 \begin{center} 
  \setlength{\unitlength}{1cm}  
 \hspace{-5cm} 
   \begin{picture}(12,8) 
    \epsfxsize=15cm 
    \epsffile{vertical.eps} 
   \end{picture} 
 \end{center} 
  \vspace*{-15cm} 
 \caption{WKB Density of the Distribution  of the eigenvalues of the Hamiltonian matrix $\hh$.} 
\label{fig1} 
\end{figure} 

Substituting  (\ref{dif-eq-odd}), (\ref{dif-eq-even}) in (\ref{wkbrho}) we 
can find the WKB density of the eigenvalues of the  Hamiltonian matrix $\hh$ (\ref{mat-modelo}).  
A straightforward calculation shows us that the conditions of Theorem \ref{teoWKB}  
are satisfied if $n$ large enough. The  expression for the $\rho^{WKB}(x)$, in both cases,  
is  large and we will only show the typical behaviour of the WKB density (see Figure 2). 
 As we can see in Figure \ref{fig1} we have that all eigenvalues are located inside the 
 support of the measure, except one, equal to $a$ in the odd case. In the picture, 
 the values $a=1,\, b=2,\, c=3$ and $d=4$ are used (from top to bottom 
 $n=10000,5000,1000,100$).  Notice that, if we look for the above picture, the eigenvalues
 of the matrix $\hh$ (the zeros of the corresponding orthogonal polynomials) are concentrated
 in the union of two intervals (with constitute the support of their orthogonality measure). \\
  
To calculate the moments $\mu_r$, we will use formula (\ref{for-mom}). Then we find, for the 
odd case, the expressions    
$$ 
\ba{l}
\mu_1=\frac{a+b}{2} + \frac{a-b}{2n+1}, \esp 
\mu_2=\frac{a^2+b^2}{2} + c^2+d^2+ \frac{a^2-b^2-c^2-d^2}{2n+1}, \esp\\
\\
\mu_3=\frac{a^3+b^3}{2}  \tres(a+b)(c^2+d^2)+ \frac{a^3-b^3-3(a+b)(c^2+d^2)}{2n+1}.
\ea  
$$ 
  
Analogously, for the even case, we have
$$ 
\ba{l}
\mu_1=\frac{a+b}{2} ,\esp 
\mu_2=\frac{a^2+b^2}{2} + c^2+d^2-\frac{d^2}{2m},\\
\\
\mu_3=\frac{a^3+b^3}{2}  \tres(a+b)(c^2+d^2)+ \frac{(a+b)(2c^2+d^2)}{2n}.
\ea  
$$
Finally, in both cases, we have the following asymptotic formulas
 $$ 
\ba{l}
\mu_1\sim\frac{a+b}{2} ,\esp \mu_2\sim \frac{a^2+b^2}{2} + c^2+d^2,\esp 
\mu_3\sim \frac{a^3+b^3}{2}  \tres(a+b)(c^2+d^2).
\ea  
$$
The dependence of the central moments via a modification of one of the interaction 
parameters $a,b,c$ or $d$  can be studied by using formulas obtained in \cite{ron-deh-paco}.
Analogous calculations can be done for 3-periodic 
Chain models. For this case computations are in progress.  
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% references  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 


 \vspace*{0.3cm} 

 \underline{\bf ACKNOWLEDGEMENTS } 
 \noindent This work was finished during a stay of the  
 first author in the  Departamento de Matem\'atica da Faculdade de Ci\^encias e  
Tecnologia da Universidade de Coimbra.  
  The research of the first two authors was partially supported  by 
 Direcci\'on General de Investigaci\'on Cient\'\i fica y T\'ecnica (DGICYT) 
 of {\sc Spain} under grant PB 93-0228-C02-01. The research of the third  
author was supported  by the grant BD976 from Junta Nacional 
 de Investiga\c c\~ao Cient\'\i fica e Tecnol\'ogica (JNICT) of {\sc Portugal}. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%                  thebibliography                    %%%%%%%%%%%%%%%%
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{\sc 
\begin{flushleft}
{{  Renato Alvarez-Nodarse} \\
Departamento de Matem\'aticas, 
Escuela Polit\'ecnica Superior,
Univ. Carlos III de Madrid,
Butarque 15, 28911 Legan\'es,
Madrid, SPAIN.} \\
{\sf E-Mail}{\small\tt : renato@dulcinea.uc3m.es }
\end{flushleft}

\begin{flushleft}
{{  Francisco Marcell\'an} \\
Departamento   de Matem\'aticas, 
Escuela Polit\'ecnica Superior,
Univ. Carlos III de Madrid,
Butarque 15, 28911 Legan\'es,
Madrid, SPAIN.} \\
{\sf E-Mail}{\small\tt : pacomarc@ing.uc3m.es }
\end{flushleft}

\begin{flushleft}
{{  Jos\'e Petronilho} \\
Departamento   de Matem\'atica, 
Faculdade de Ci\^encias e Tecnologia,
Univ. Coimbra,
Apartado 3008,
Coimbra, PORTUGAL.} \\
{\sf E-Mail}{\small \tt : josep@mat.uc.pt }
\end{flushleft} }

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