%%
%%  This is AmSLaTeX    lor2004.tex       mar 2004
%%
%% Here is a character listing to check to be sure that no
%% unwanted translations took place within the bowels of the net.
%% Upper case letters: ABCDEFGHIJKLMNOPQRSTUVWXYZ
%% Lower case letters: abcdefghijklmnopqrstuvwxyz
%% round parentheses, square brackets, curly braces: ()  []  {}
%% Exclaim, at, sharp, dollar, percent: ! @ # $ %
%% Caret, ampersand, star, underscore, hyphen: ^ & * _ -
%% vertical bar, backslash, tilde, backprime, plus: | \ ~ ` +
%% plus, equal, prime, quote, colon: + = ' " :
%% less than, greater than, slash, question, comma: < > / ? ,
%% period, semicolon: . ;
%%
%%
\documentclass[10pt,reqno,oneside,final]{amsart}

%page size
\hsize=7.7677truein\vsize=10.5truein

% LaTeX's page layout parameters   Pedro Quaresma de Almeida
%                                   1995

\textwidth=7.5in
\textheight=9.0in
\oddsidemargin=-.6in
\evensidemargin=-.6in
\headheight=0.5in
\topmargin=-.05in
\headsep=0.25in
\voffset=-0.5in

\input epsf.tex \epsfverbosetrue

\makeatletter
%\def\ddots{\mathinner{\mkern1mu\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern2mu
%    \raise4\p@\hbox{.}\mkern2mu\raise\p@\hbox{.}\mkern1mu}}
\def\secddots{\mathinner{\mkern1mu\raise\p@\vbox{\kern7\p@\hbox{.}}\mkern2mu
    \raise4\p@\hbox{.}\mkern2mu\raise7\p@\hbox{.}\mkern1mu}}
\makeatother

\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{array}
\usepackage{cite}
\usepackage{picins}
\usepackage{fancyhdr}
%\usepackage{pslatex}\renewcommand{\boldsymbol}{\pmb}

\def\mylmark{\ }
\markboth{}{} \markright{}

\pagestyle{fancy}
\lhead{\rmfamily \textsf{Lorentz weights 2004\ }--\ \textbf{\thesection}\ --\ \mylmark\  --} \chead{} \rhead{\rmfamily\thepage}
\lfoot{}\cfoot{}\rfoot{}



%\renewcommand{\chaptername}{CHAPITRE}
%\renewcommand{\contentsname}{Table (provisoire) des mati\`eres.\ \today}
%\renewcommand{\contentsname}{Table  des mati\`eres.\ }

\usepackage{pst-plot}

% tables des matieres avec ... : extrait de amsbook.cls avec dotfill au
%   de hfil
\makeatletter
\def\@tocline#1#2#3#4#5#6#7{\relax
  \ifnum #1>\c@tocdepth % then omit
  \else
    \par \addpenalty\@secpenalty\addvspace{#2}%
    \begingroup \hyphenpenalty\@M
    \@ifempty{#4}{%
      \@tempdima\csname r@tocindent\number#1\endcsname\relax
    }{%
      \@tempdima#4\relax
    }%
    \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax
    \rightskip\@pnumwidth plus1em \parfillskip-\@pnumwidth
    #5\leavevmode\hskip-\@tempdima #6\relax
    \ \ \dotfill\hbox to\@pnumwidth{\@tocpagenum{#7}}\par
    \nobreak
    \endgroup
  \fi}

% police de chapitre
\def\@makechapterhead#1{\global\topskip 7.5pc\relax
  \begingroup
  \fontsize{\@xivpt}{18}\bfseries\centering
    \ifnum\c@secnumdepth>\m@ne
      \leavevmode \hskip-\leftskip
      \rlap{\vbox to\z@{\vss
%         \centerline{\normalsize\bfseries
          \centerline{\Large    \bfseries
                        \@xp{\chaptername}\enspace\thechapter}
          \vskip 3pc}}\hskip\leftskip\fi
     #1\par \endgroup
  \skip@34\p@ \advance\skip@-\normalbaselineskip
  \vskip\skip@ }
%\newcommand{\@chapapp}{\chaptername}

\def\section{\@startsection{section}{1}%
  \z@{.7\linespacing\@plus\linespacing}{.5\linespacing}%
  {\normalfont\bfseries\centering}}

\makeatother

\usepackage[dvips]{hyperref}


\def\M{\boldsymbol{M}}
\def\d{\text{d}}

\begin{document}

%\textheight=9.00in   %9.5in
\typeout{ oddsidemargin  \the\oddsidemargin}



\begin{center}
         {\Large Formulas for recurrence coefficients of
              orthogonal polynomials related to Maxwellian
             and Lorentzian weights.}

         \bigskip

          A.P. Magnus and V. Pierrard

Universit\'e Catholique   de Louvain-La-Neuve, 2 Chemin du Cyclotron, B-1348
Louvain-La-Neuve, Belgium, and\\
Institut d'A\'eronomie Spatiale de Belgique, 3 av. Circulaire, B-1180
Brussels.

 magnus@anma.ucl.ac.be ; Viviane.Pierrard@oma.be

         \bigskip

This version: \today \qquad  (incomplete and unfinished)
\end{center}

\section{Introduction.}

We call ``Maxwellian polynomials'' (or ``speed polynomials'' when
$\alpha=2$ below),
and ``Lorentzian polynomials'' a set of polynomials $\pi_i$
 of degree $i$ with respect to the weight functions
\begin{equation}  \label{max}
   w(t) = t^\alpha \exp(-t^2)
\end{equation}
and
\begin{equation}  \label{lor}
   w(t) = t^\alpha (1+t^2)^{-\beta}
\end{equation}
on the positive half-line $0\leqslant t<\infty$.

\section{Recurrence coefficients.}

We proceed with general identities which will be needed here, and
which can be found in any general textbook on orthogonal
polynomials, such as \cite{chi,gaut04,szeg}

We consider a set of polynomials $\pi_k$ of degree $k$ for
 each $k=0,1,\ldots$ and all of which are mutually orthogonal over
 the weight
function $w$:
\begin{eqnarray}
\int_a^b w(t) \pi_k(y) \pi_m(t) \d t=0  \hspace{1.cm}{\rm if}\ k\neq m.
\end{eqnarray}
Monic orthogonal polynomials can be generated from the general recurrence
relation
\begin{eqnarray}
\pi_{-1}(x)&=&0\\
\pi_0(x)&=&1\\
\pi_{k+1}(x)&=&(x-\alpha_k)\pi_k(x)-\beta_k\pi_{k-1}(x)\hspace{1.cm} k=0,1,...
 \label{recur}
\end{eqnarray}
where $\alpha_k$ and $\beta_k$ are defined by:
\begin{eqnarray}
\alpha_k&=&\frac{\int_a^b t \pi^2_k(t) w(t) \d t}{\int_a^b
\pi^2_k(t)w(t)\d t}\label{ak}\hspace{1.cm} k=0,1,...\\
\beta_k&=&\frac{\int_a^b \pi^2_k(t) w(t) \d t}{\int_a^b
\pi^2_{k-1}(t)w(t)\d t}\hspace{1.cm} k=1,2,...
\label{bi}
\end{eqnarray}
%The  orthonormal polynomials are given by:
%\begin{equation}
% \mathcal{Q}_i(y)=\frac{Q_i(y)}{\sqrt{\int_0^c Q^2_i(y)W(y)dy}}.
%\end{equation}

With these definitions, the first coefficients can be calculated
analytically:

etc.

\emph{The fast and accurate computation of these recurrence
coefficients for the special families of weights \eqref{max}
and \eqref{lor} is the subject-matter of the present paper.
}

Intermediate polynomials: if $a>-\infty$, one considers
 $\hat{\pi}_k$ orthogonal on
 $(-\sqrt{b-a},\sqrt{b-a})$
with respect to the {\bf even} weight function $\hat{w}(u)=
|u|W(a+u^2)$,
so that
\begin{eqnarray}
\int_{-\sqrt{b-a}}^{\sqrt{b-a}} \hat{\pi}_{2k}(u)\hat{\pi}_{2m}(u)
 \hat{w}(t)\;\d t =
2\int_0^{\sqrt{b-a}} \hat{\pi}_{2k}(u)\hat{\pi}_{2m}(u) u w(a+u^2)\;\d t =
\int_a^b \hat{\pi}_{2k}(\sqrt{t-a})\hat{\pi}_{2m}(\sqrt{t-a}) w(t)\;\d t ,
\end{eqnarray}
by $u=\sqrt{t-a}$, and $\hat{\pi}_{2k}(u)= \pi_k(a+u^2)$ follows( see,
for instance, \cite[\S~1.18]{gaut04}).

Consider now the recurrence relation for the $\hat{\pi}_k$'s:
\begin{equation}
\hat{\pi}_{-1}(u)=0\; \quad \hat{\pi}_0(u)=1\; \quad%
\hat{\pi}_{k+1}(u)=u\hat{\pi}_k(u)-\gamma_k\hat{\pi}_{k-1}(u)\qquad k=0,1,...
 \label{recurg}
\end{equation}
Then,
\begin{eqnarray}
\hat{\pi}_{2k+2}(u)&=&u[u\hat{\pi}_{2k}(t)-\gamma_{2i}\hat{\pi}_{2k-1}]-
\gamma_{2k+1}         \hat{\pi}_{2k}(t)   \nonumber \\
  &=& u^2 \hat{\pi}_{2k}(u) -\gamma_{2k} [\hat{\pi}_{2k}(u)+
  \gamma_{2k-1}\hat{\pi}_{2k-2}(u)]
      -\gamma_{2k+1} \hat{\pi}_{2k}(u) \nonumber
\end{eqnarray}
whence, from $\hat{\pi}_{2k}(u)=\pi_k(t)$ and $u^2=t-a$,
\begin{eqnarray}
\alpha_k = a+\gamma_{2k}+\gamma_{2k+1}\;,\qquad\qquad
\beta_k = \gamma_{2k-1} \gamma_{2k}\;, \qquad\qquad k=0,1,\ldots
 \label{gamma}
\end{eqnarray}
(where one has to start with $\gamma_0=0$).

We now have to deal only with the single sequence $\gamma_1,
\gamma_2,\ldots$

As in (\ref{bi}),
\begin{eqnarray}
\gamma_k=\frac{\int_{-\sqrt{b-a}}^{\sqrt{b-a}}
 \hat{\pi}^2_k(u) \hat{w}(u) \d u}
      {\int_{-\sqrt{b-a}}^{\sqrt{b-a}}
       \hat{\pi}^2_{k-1}(u)\hat{w}(u)\d u}
  = \frac{ \|\hat{\pi}_k\|^2}{\|\hat{\pi}_{k-1}\|^2}
       \hspace{1.cm} i=1,2,....
\label{gami}
\end{eqnarray}

We will often need expansions of products $u^p \hat{\pi}_k(u)$ in the
basis $\{\hat{\pi}_0, \hat{\pi}_1,\ldots\}$. We only have to iterate (\ref{recurg})
in the form $u\hat{\pi}_k(u)= \hat{\pi}_{k+1}(u)+\gamma_k \hat{\pi}_{k-1}(u)$:

\begin{eqnarray}     \label{matri}
 u {\bf \hat{\pi}}(u) = \M\, {\bf \hat{\pi}}(u)\;: \qquad
u\,\left[\begin{array}{c} \hat{\pi}_0(u) \\ \hat{\pi}_1(u) \\ \hat{\pi}_2(u) \\  \vdots
           \end{array}\right] =
\left[\begin{array}{ccccc} 0    &    1     &      &    &  \\
                         \gamma_1&  0     &   1  &     &   \\
                                 &\gamma_2&  0   &   1 &   \\
                                 &        &\ddots&\ddots&\ddots
           \end{array}\right]
 \left[\begin{array}{c} \hat{\pi}_0(u) \\ \hat{\pi}_1(u) \\ \hat{\pi}_2(u) \\  \vdots
           \end{array}\right]
\end{eqnarray}
then, $u^p{\bf \hat{\pi}}(u)=\M^p {\bf \hat{\pi}}(u)$, so that
 $\M^p_{i,j}$ is
the coefficient of $\hat{\pi}_j(u)$ in the expansion of $u^p \hat{\pi}_i(u)$
(the indexes start at 0).

First instances:
\begin{eqnarray}     \label{matri2}
 \begin{array}{cccccccc}
       &  \hat{\pi}_{i-3}(u)   &  \hat{\pi}_{i-2}(u)   &  \hat{\pi}_{i-1}(u)   &  \hat{\pi}_i(u)
       &  \hat{\pi}_{i+1}(u)   &  \hat{\pi}_{i+2}(u)   &  \hat{\pi}_{i+3}(u)   \\
u\hat{\pi}_i(u)&               &                &  \gamma_i    &    0
       &    1          &               &               \\
u^2\hat{\pi}_i(u)&             &\gamma_i\gamma_{i-1}&  0    & \gamma_i+\gamma_{i+1}
       &    0          &     1         &               \\
u^3\hat{\pi}_i(u)&\gamma_i\gamma_{i-1}\gamma_{i-2}&  0    &
\gamma_i(\gamma_{i-1}+\gamma_i+\gamma_{i+1})& 0 &
\gamma_{i}+\gamma_{i+1}+\gamma_{i+2} & 0 &1
 \end{array}
\end{eqnarray}

For $u^4$,
$u^4\hat{\pi}_i(u)=\gamma_i\gamma_{i-1}\gamma_{i-2}\gamma_{i-3}
\hat{\pi}_{i-4}(u)+
\gamma_i\gamma_{i-1}(\gamma_{i-2}+\gamma_{i-1}+\gamma_i+\gamma_{i+1})
\hat{\pi}_{i-2}(u)+
+(\gamma_i\gamma_{i-1} +\gamma_i^2 +2\gamma_i\gamma_{i+1}
 +\gamma_{i+1}^2 +\gamma_{i+1}\gamma_{i+2})\hat{\pi}_i(u)
+(\gamma_i+\gamma_{i+1}+\gamma_{i+2}+\gamma_{i+3})
\hat{\pi}_{i+2}(u)+
\hat{\pi}_{i+4}(u)$

\section{Differential relations.}

\subsection{Orthogonal polynomials satisfying differential
  relations and equations}  \   \\

We now come to a special class of weight functions allowing
remarkable relations for the recurrence coefficients:

\noindent\textbf{Lemma.}\textsl{If the logarithmic derivative
$w'/w$ is a rational function, the recurrence coefficients
satisfy exactly computable equations  }
\begin{equation*}
 F_k(\alpha_0,\beta_1,\dotsc, \alpha_{k+d-1},\beta_{k+d})=0,  \qquad%
 G_k(\alpha_0,\beta_1,\dotsc, \alpha_{k+d-1},\beta_{k+d},\alpha_{k+d})=0,
\end{equation*}
\textsl{for $k=1,2,\dotsc$, where $d$ depends on the degree of the rational
function $w'/w$. Moreover, the orthogonal polynomials satisfy
differential relations and equations of the form
\begin{equation*}
P\pi'_k = Q_k \pi_k+R_k \pi_{k-1},  \qquad%
PR_k \pi''_k +U_k \pi'_k +V_k \pi_k=0,
\end{equation*}  where $P,Q_k$, $R_k$, $U_k$, and $V_k$ are
polynomials of fixed degree.}

This statement has been discovered and rediscovered in various
forms, see \cite{MagP,Sho}. Most authors are interested in the
differential formulas for $\pi_k$, but the computationnaly
interesting items are the $F_k$ and $G_k$'s.

\subsection{The making of the equations}  \   \\

We give here only a part of the proof for the even weight
$\hat{w}(u)=|u|w(a+u^2)$ on $|u|< c=\sqrt{b-a}$. Suppose
we have $\hat{w}'(u)/\hat{w}(u) = q(u)/p(u)$, where we also
manage to have $\lim p(u)\hat{w}(u)=0$ when $u\to \pm c$.
 Then, by integration by parts,

\begin{equation}\begin{split}   \label{ipart}
\int_{-c}^c p(u)\hat{\pi}_{k-1}(t)\, \hat{\pi}'_k(u)\,\hat{w}(u)\,\d u &=
-\int_{-c}^c  \hat{\pi}_k(u)\,[p(u)\hat{\pi}_{k-1}(u)\hat{w}(u)]'\,\d u\\
&=-\int_{-c}^c  \hat{\pi}_k(u)\,p'(u)\hat{\pi}_{k-1}(u)\hat{w}(u)\,\d u\\
& -\int_{-c}^c  \hat{\pi}_k(u)\,p(u)\hat{\pi}'_{k-1}(u)\hat{w}(u)\,\d u\\
& -\int_{-c}^c  \hat{\pi}_k(u)\,\hat{\pi}_{k-1}(u)\,p(u)\hat{w}'(u)\,\d u
\end{split}\end{equation}

The first and the third of these latter integrals involve the
product of $\hat{\pi}_k$
 and the polynomial of fixed degree
$p'+q$ (after replacement of $p\hat{w}'$ by $q\hat{w}$). By the rules
of \eqref{matri2}, the product is a linear combination of, say,
$\hat{\pi}_{k+d}$, $\hat{\pi}_{k+d-1},\dotsc$ with coefficients
$[(p'+q)(\M)]_{k,j}, j=k+d,k+d-1,\dotsc$
which are simple polynomials in $\gamma_{k+d},\dotsc, \gamma_{k-d}$.
Then, by orthogonality of the $\hat{\pi}$'s with respect to
$\hat{w}$, the value of the two integrals comes out as the
coefficient of $\hat{\pi}_{k-1}$ times
$\|\hat{\pi}_{k-1}\|^2$.

The left-hand side and the second of the three latter integrals
are estimated in a similar way, after having written the derivative
of a $\hat{\pi}$ polynomial in its own basis:

\begin{equation}\label{deriv}
  \hat{\pi}'_k(u) = k \hat{\pi}_{k-1}(u)
                    +\delta_k \hat{\pi}_{k-3}(u)
                    +\epsilon_k \hat{\pi}_{k-5}(u)  +\dotsb,
\end{equation}
so that the left-hand side of \eqref{ipart} is
\begin{equation*}
 k [p(\M)]_{k-1,k-1} \|\hat{\pi}_{k-1}\|^2
 +\delta_k [p(\M)]_{k-3,k-1} \|\hat{\pi}_{k-1}\|^2
 +\epsilon_k [p(\M)]_{k-5,k-1} \|\hat{\pi}_{k-1}\|^2  +\dotsb
\end{equation*}
and the right-hand side is
\begin{equation*}
 -(k-1) [p(\M)]_{k-2,k} \|\hat{\pi}_k\|^2
 -\delta_{k-1} [p(\M)]_{k-4,k} \|\hat{\pi}_k\|^2
 -\epsilon_{k-1} [p(\M)]_{k-6,k} \|\hat{\pi}_k\|^2  +\dotsb
 -[(p'+q)(\M)]_{k,k-1}  \|\hat{\pi}_{k-1}\|^2,
\end{equation*}
and the final equation is found after dividing by $\|\hat{\pi}_{k-1}\|^2$,
reminding that
 $\|\hat{\pi}_k\|^2/\|\hat{\pi}_{k-1}\|^2=\gamma_k$,
\begin{equation} \label{eqF}     \begin{split}
F_k(\gamma_1,\dotsc, \gamma_{k+d})&=
 -[(p'+q)(\M)]_{k,k-1}
- k [p(\M)]_{k-1,k-1}
 -\delta_k [p(\M)]_{k-3,k-1}
 -\epsilon_k [p(\M)]_{k-5,k-1} -\dotsb    \\
 &
 -\gamma_k\{ (k-1) [p(\M)]_{k-2,k}
 +\delta_{k-1} [p(\M)]_{k-4,k}
 +\dotsb\} =0,
\end{split}\end{equation}
for $k=1,2,\dotsc$  One needs a   number of terms of the expansion
in $\delta, \epsilon,\dotsc$ which depends on the width of the
bandmatrix $p(\M)$.

The coefficients  $\delta_k$, $\epsilon_k,\dotsc$ in \eqref{deriv}
 are polynomials in the $\gamma$'s too:
from $\displaystyle{ \dfrac{ \hat{\pi}_k(u) }
                           { \hat{\pi}_{k-1}(u) } =
   u-\dfrac{  \gamma_{k-1} }{u -\dfrac{\gamma_{k-2}}{u-\dotsb} }   }$
$= u -\gamma_{k-1}u^{-1} -\gamma_{k-1}\gamma_{k-2} u^{-3}
  +O(u^{-5})  $,

\begin{equation*}
\dfrac{\hat{\pi}'_k(u)}
      {\hat{\pi}_k(u)}
-\dfrac{\hat{\pi}'_{k-1}(u)}
      {\hat{\pi}_{k-1}(u)}
=\dfrac{\d }{\d u}\log\left(\dfrac{\hat{\pi}_k(u)}
                                 {\hat{\pi}_{k-1}(u)}  \right) =
u^{-1} +2\gamma_{k-1}u^{-3} +[2\gamma_{k-1}^2+4\gamma_{k-1}
  \gamma_{k-2}]u^{-5}+O(u^{-7})
\end{equation*}
whence, by summing,
\begin{equation}    \label{ppp}
\dfrac{\hat{\pi}'_k(u)}
      {\hat{\pi}_k(u)}  =
k\dfrac{\hat{\pi}_{k-1}}{\hat{\pi}_k}
+\delta_k\dfrac{\hat{\pi}_{k-3}}{\hat{\pi}_k}
+\epsilon_k\dfrac{\hat{\pi}_{k-5}}{\hat{\pi}_k}
+O(u^{-7})=
ku^{-1} +2\sum_2^{k-1}\gamma_i u^{-3} +
  \sum_2^{k-1}[2\gamma_i^2+4\gamma_i \gamma_{i-2}]u^{-5}+O(u^{-7})
\end{equation}
Comparing the coefficients of $u^{-3}$ and $u^{-5}$ in the expansions:
\begin{equation} \label{deleps}
 \delta_k = 2\sum_1^{k-1} \gamma_i -k\gamma_{k-1}\,\qquad%
 \epsilon_k = 2\sum_1^{k-1} [\gamma_i^2+2\gamma_i \gamma_{i-1}]
  -k[\gamma_{k-1}^2+\gamma_{k-1}\gamma_{k-2}]
  -\delta_k[\gamma_{k-1}+\gamma_{k-2}+\gamma_{k-3}]
\end{equation}

\subsection{Exercise 1: Laguerre and Hermite polynomials} \   \\

\begin{equation*}
  w(t)= t^\alpha \exp(-t) \text{\ on\ } (0,\infty): \qquad%
  \hat{w}(u)= u^{2\alpha+1} \exp(-u^2) \text{\ on\ } (-\infty,\infty): \qquad%
\dfrac{\hat{w}'(u)}{\hat{w}(u)} =
 \dfrac{2\alpha+1}{u} -2u
\end{equation*}

The sensible choice seems to be $p(u)=u^2$ ($p$ must be an even
polynomial), and $q(u)=(2\alpha+1)u-2u^3$. As $\M^2$ is a
five-diagonal matrix, the nonzero terms of \eqref{eqF} are
\begin{equation*}
-(2\alpha+3)[\M]_{k,k-1}
+2[\M^3]_{k,k-1}
-k[\M^2]_{k-1,k-1}
-\delta_k[\M^2]_{k-3,k-1}
-(k-1)\gamma_k[\M^2]_{k-2,k}=0,
\end{equation*}
or
\begin{equation*}
-(2\alpha+3)\gamma_k
+2\gamma_k(\gamma_{k-1}+\gamma_k+\gamma_{k+1})
-k(\gamma_{k-1}+\gamma_k)
-\delta_k
-(k-1)\gamma_k =
-(2\alpha+3)\gamma_k
+2\gamma_k(\gamma_{k-1}+\gamma_k+\gamma_{k+1})
-(2k-1)\gamma_k -2\sum^{k-1}\gamma_i=0
\end{equation*}

We keep the degree of $p$ as low as possible, so to avoid
big bandmatrices in \eqref{eqF}: with
$p=1$,  $q$ is not a polynomial, but a polynomial divided by $u$,
and \eqref{eqF} has a strange term with $\M^{-1}$:
\begin{equation*}
-(2\alpha+1)[\M^{-1}]_{k,k-1}
+2[\M]_{k,k-1}
-k =0.
\end{equation*}
But $[f(\M)]_{i,j}$ is a shorthand for the integral
of $f(u)\hat{\pi}_i(u)\hat{\pi}_j(u) \hat{w}(u)$ divided by
$\|\hat{\pi}_j\|^2$. There is nothing wrong with
of $u^{-1}\hat{\pi}_k(u)\hat{\pi}_{k-1}(u)$, as
 $\hat{\pi}_k\hat{\pi}_{k-1}$ is an \emph{odd} polynomial! The result
is 0 if $k$ is even, and 1 if $k$ is odd. One then gets immediately
\begin{equation*}
\gamma_k = \dfrac{ k +(\alpha+1/2)[1-(-1)^k]}{2}.
\end{equation*}


\subsection{Exercise 2: Laguerre and Hermite weights on bounded
intervals} \   \\

\begin{equation*}
  w(t)= t^\alpha \exp(-t) \text{\ on\ } (0,c): \qquad%
  \hat{w}(u)= u^{2\alpha+1} \exp(-u^2) \text{\ on\ }
   (-\sqrt{c},\sqrt{c}): \qquad%
\dfrac{\hat{w}'(u)}{\hat{w}(u)} =
 \dfrac{2\alpha+1}{u} -2u
\end{equation*}

We must now take  be $p(u)=c-u^2$,
 so,  $q(u)=(2\alpha+1)c/u-(2c+2\alpha+1)u+2u^3$, and
 \eqref{eqF} becomes
\begin{equation*}
-(\alpha+1/2)c[1-(-1)^k]
+(2c+2\alpha+3)[\M]_{k,k-1}
-2[\M^3]_{k,k-1}  -kc
+k[\M^2]_{k-1,k-1}
+\delta_k[\M^2]_{k-3,k-1}
+(k-1)\gamma_k[\M^2]_{k-2,k}=0,
\end{equation*}
or
\begin{equation*}
-(2\alpha+3)\gamma_k
+2\gamma_k(\gamma_{k-1}+\gamma_k+\gamma_{k+1})
-k(\gamma_{k-1}+\gamma_k)
-\delta_k
-(k-1)\gamma_k =
-(2\alpha+3)\gamma_k
+2\gamma_k(\gamma_{k-1}+\gamma_k+\gamma_{k+1})
-(2k-1)\gamma_k -2\sum^{k-1}\gamma_i=0
\end{equation*}


\section{Maxwellian weight.}

From \eqref{max}, one has
\begin{eqnarray}
w(t) = t^\alpha\exp{(-t^2)}:\qquad
\hat{w}(u)=|u|^{2\alpha+1}\exp(-u^4):\qquad
\dfrac{\hat{w}'(u)}{\hat{w}(u)} = \dfrac{2\alpha+1}{u} -4u^3
      \label{maxw}
\end{eqnarray}
on the whole real line for $u$.
\begin{equation*}
F_k=  -(2\alpha+1)[\M^{-1}]_{k,k-1} +4[\M^3]_{k,k-1}-k=0.
\end{equation*}

the sought relation is
\begin{equation}
F_k = -(\alpha+1/2)[1-(-1)^k] +4\gamma_k
(\gamma_{k+1}+\gamma_k+\gamma_{k-1}) -k=0,\qquad k=1,2,\dotsc
  \label{maxwg}
\end{equation}

by various authors \cite[...]{Bes,Freud,LeQ,Sho} is established:

\medskip

This remarkably simple relation seems to allow the computation of
any sequence $\{\gamma_1,\ldots, \gamma_N\}$ from the knowledge of
the single $\gamma_1=\alpha_0$! However, the obvious repetition
of $\gamma_{i+1}= [i+(2A+1)(1-(1)^i)/2 -\gamma_i-\gamma_{i-1}]/
(4B\gamma_i)$ soon turns into a numerical nightmare. Any numerical error
in $\gamma_1$ is strongly amplified in the subsequent $\gamma_i$'s.

\medskip

Instead of considering (\ref{maxwg}) as an initial value problem, we
have to consider it as a nonlinear boundary value problem for
$\gamma_1,\ldots, $, given $\gamma_0=0$, and knowing that $\gamma_i>0$
for $i=1,2,\ldots$. A numerically valuable use of (\ref{maxwg})
consists in correcting a whole positive sequence
$\gamma_{1,{\rm old}},\ldots,$ by seeing each instance of
(\ref{maxwg}) as an algebraic equation for $\gamma_i$:
\begin{eqnarray}
\gamma_{i,{\rm new}} =
-\frac{\gamma_{i+1,{\rm old}}+\gamma_{i-1,{\rm old}}}{2} +\sqrt{\left(
\frac{\gamma_{i+1,{\rm old}}+\gamma_{i-1,{\rm old}}}{2}\right)^2
+\frac{ i+(2A+1)(1-(-1)^i)/2}{4B}}\;,\ \ \ i=1,2,\ldots    \label{gamiter1}
\end{eqnarray}
which sends positive sequences on positive sequences, may be shown
to be contractive, and has interesting by-products, such as to allow
a formal proof of the asymptotic behaviour
\begin{eqnarray}
\gamma_i = \sqrt{\frac{i}{12B}} +o(i^{1/2)} )
\end{eqnarray}
when $i\to\infty$, \cite{Freud,Nevai}. One sees then how to build
a satisfactory finite sequence $\gamma_1,\ldots, \gamma_N$ by
putting the boundary value $\gamma_{N+1} = \sqrt{(N+1)/(12B)}$ for
a large $N$ \cite{ClS}.


Much more efficient Newton-Raphson iteration: see \cite{LeQ}

{\small
\begin{verbatim}

 1: x**  2. exp(-  1.x**2) on (0,infty)
  max err=  0.711324865
  how many iterations?  0=stop iterations
7
  max err=  0.834080302
  max err=  0.134694001
  max err=  0.005634623
  max err=  1.12011135E-05
  max err=  4.45810495E-11
  max err=  2.0275448E-14
  max err=  2.0275448E-14
  how many iterations?  0=stop iterations
0
  n      gamma 2n          gamma 2n+1          alpha n          beta n

  0    0.000000000000   1.128379167096   1.128379167096   0.000000000000
  1    0.200961221084   1.158701818690   1.359663039774   0.226760455265
  2    0.366406573426   1.204100286235   1.570506859662   0.424555963009
  3    0.505732161771   1.256164404453   1.761896566224   0.608952240747
  4    0.626325858616   1.310735711057   1.937061569673   0.786768249182
  5    0.733194509013   1.365806172210   2.099000681223   0.961024226114
  6    0.829672567872   1.420405545084   2.250078112955   1.133171914113
  7    0.918031174969   1.474069974087   2.392101149057   1.303976571486
  8    0.999868084355   1.526589673754   2.526457758109   1.473875521196
  9    1.076344025868   1.577885714852   2.654229740719   1.643135675296
 10    1.148327024657   1.627947610843   2.776274635501   1.811928808185
 20    1.718530821304   2.070307121180   3.788837942485   3.488765218913
 30    2.146704699404   2.437506231270   4.584210930674   5.159052852061
 40    2.503790613727   2.757202157410   5.260992771138   6.827575680737
 50    2.816370577692   3.043877061716   5.860247639408   8.495370970381
 60    3.097760669007   3.305963109633   6.403723778640  10.162796343342
 70    3.355740791483   3.548842545743   6.904583337226  11.830008047078
 80    3.595318011570   3.776189798576   7.371507810146  13.497085197754
 90    3.819937524891   3.990642533269   7.810580058160  15.164072144673
100    4.032089039769   4.194168861795   8.226257901563  16.830995676797

\end{verbatim}
}


\section{Lorentzian weight.}

Now, from \eqref{lor}, one has
\begin{eqnarray}
w(t) = t^\alpha (1+t^2)^{-\beta}:\qquad
\hat{w}(u)=|u|^{2\alpha+1} (1+u^4)^{-\beta}:\qquad
\dfrac{\hat{w}'(u)}{\hat{w}(u)} = \dfrac{2\alpha+1}{u}
 -(4\beta-2\alpha-2)u^3
      \label{lorw}
\end{eqnarray}
on the whole real line for $u$.

So,
\begin{equation*}
 p(u)=1+u^4,\qquad q(u)=(2\alpha+1)\left( \dfrac{1}{u}+u^3\right)
 -4\beta u^3,\qquad p'(u)+q(u)= \dfrac{2\alpha+1}{u}
  -(4\beta-2\alpha-5)u^3.
\end{equation*}

There is nothing wrong in considering the Lorentzian weight
$\hat{w}(u) = |u|^{2\alpha+1}\, (1+u^4)^{-\beta}\;,\  \  \
\alpha>-1, \beta>0$,
on $(-\infty,\infty)$, as long as
 the integrals in (\ref{gami})
only involve functions decreasing faster than $|u|^{-1}$ when
$|u|\to\infty$. So, $\gamma_k$ still exists if
$2k+2\alpha+1 -4\beta <-1$, or

\begin{eqnarray}     \label{ilimit}
    k < 2\beta-1-\alpha
\end{eqnarray}


Equation \eqref{eqF} is now

\begin{equation*} \begin{split}
F_k&=-(\alpha+1/2)[1-(-1)^k] +(4\beta-2\alpha-5)[\M^3]_{k,k-1}
   -k -k[\M^4]_{k-1,k-1} -\delta_k[\M^4]_{k-3,k-1} -\epsilon_k  \\
& -\gamma_k\{ (k-1)[\M^4]_{k-2,k} +\delta_{k-1}\}=0
\end{split}\end{equation*}
or
\begin{equation}  \label{lorFk}
 F_k = 2(2\beta -\alpha -2-k)\gamma_k\gamma_{k+1} +\rho_k =0,
\end{equation}

where $\rho_k$ is the sum of terms in $F_k$ not involving
$\gamma_{k+1}$:
\begin{equation}\label{lorrhok}\begin{split}
 \rho_k &=
-(\alpha+1/2)[1-(-1)^k] +(4\beta-2\alpha-2)\gamma_k(
  \gamma_{k-1}+\gamma_k)     \\
& -k[1+\gamma_{k-1}\gamma_{k-2}+\gamma_{k-1}^2
    +2\gamma_{k-1}\gamma_k +\gamma_k^2]
 -\delta_k(\gamma_{k-3}
          +\gamma_{k-2}
          +\gamma_{k-1}
          +\gamma_k)
  -\epsilon_k    \\
 &  -(k-1)\gamma_k(\gamma_{k-2}+\gamma_{k-1}+\gamma_k)
  -\delta_{k-1}\gamma_k    \\
 &=
-(\alpha+1/2)[1-(-1)^k] +2(2\beta-\alpha-2-k)\gamma_k
  (\gamma_{k-1}+\gamma_k) \\
  & -k-4\gamma_k\sum^{k-2}\gamma_i  -2\gamma_k\gamma_{k-1}
      -2\sum^{k-1}[\gamma_i^2 +2\gamma_i\gamma_{i-1}],
\end{split}\end{equation}


which we may as well compute, while
$   k < 2\beta-\alpha-2  $ , directly for $\gamma_{k+1}$, although in
high precision:
\begin{equation*}
 \gamma_{k+1} = -\dfrac{\rho_k}{2(2\beta -\alpha -2-k)\gamma_k},
\end{equation*}
but numerical instability soon settles in if $\beta$ is large.


%{\Small
%\textbf{Lemma.}\textsl{When $2\beta-\alpha-2$ tends toward an
%integer $N$, one has}
%\begin{equation} \label{Nlim}
% \lim_{\substack{\varepsilon\to 0 \\ \varepsilon=2\beta-\alpha-2-N>0}}
%  \rho_N\;\|\hat{pi}_{N-1}\|^2  = -2.
%\end{equation}
%
%Indeed, while $2\beta-\alpha-2-N=\varepsilon>0$, we may
%still evaluate $\displaystyle{  \|\hat{\pi}_{N+1}\|^2 =
% \int_{-\infty}^\infty \hat{\pi}^2_{N+1}(u) |u|^{2\alpha+1}
%  (1+u^4)^{-\beta}\,\d u }$, which is mainly the integral of
%$|u|^{2N+2+2\alpha+1-4\beta} =|u|^{-1-2\varepsilon} $ on a large
%interval, so,
%\begin{equation*}
% \|\hat{\pi}_{N+1}\|^2 = \|\hat{\pi}_0\|^2 \gamma_1 \dotsc \gamma_N
% \gamma_{N+1} = \dfrac{1}{\varepsilon} +O(1),
%\end{equation*}
%and $0=F_N = 2\varepsilon \gamma_N\gamma_{N+1} +\rho_N$: \qed
%}

Sensible way is again to compute the $\gamma_n$'s from the two
boundary values $\gamma_0=0$, and some $\gamma_N$ with $N$ not far
from $2\beta-\alpha$, should such a value be available\dots

This can be done for special values of $\alpha$ and $\beta$:

\textbf{Lemma.}\textsl{When $\beta$ is an integer, and when $2\alpha$
is an odd integer, one has}
\begin{equation} \label{halfin}
 \left. \begin{matrix}
  \gamma_{2\beta-2\alpha-2}= \dfrac{4\beta^2 +4\alpha^2+12\alpha+7}
                             {8(\alpha+1)(\alpha+2)},  \\
  \gamma_{2\beta-2\alpha-1}= \dfrac{4\beta^2 -(2\alpha+1)^2}
                             {8\alpha(\alpha+1)}, \\
  \gamma_{2\beta-2\alpha}= \dfrac{4\beta^2 +4\alpha^2-4\alpha-1}
                             {8\alpha(\alpha-1)}.
  \end{matrix} \right\}
\end{equation}
%\begin{center} \begin{tabular}{c||c|c||c|c}
%           $\alpha$ &      $n$  &   $\gamma_n$ & $n$ & $\gamma_n$  \\
%\hline     $-1/2$   & $2\beta-2$ &
%  $2(\beta^2-1)(2\beta^2+7)/[15(2\beta^2+1)]$ &
%  $2\beta-1$  &  $(2\beta^2+1)/3$ \\
%           $1/2$   &  $2\beta-3$  &  $(2\beta^2+7)/15$
%             & $2\beta-2$  & $2(\beta^2-1)/3$ \\
%           $3/2$ &  $ 2\beta-4$ & $2(\beta^2-4)/15$
%             & $2\beta-3$  & $(2\beta^2+1)/3$   \\
%           $5/2$ &  $ 2\beta-5$ & $(2\beta^2+7)/15$
%             & $2\beta-4$  & $2(\beta^2-4)(2\beta^2+1)/
%             [3(2\beta^2+7)]$
%\end{tabular}  \end{center}

Indeed,
\begin{enumerate} \item
 when $\alpha=-1/2$, we have to consider orthogonal polynomials
with respect to
$\hat{w}(u)= (1+u^4)^{-\beta}$ on $(-\infty,\infty)$. The two last
ones are
$$
\hat{\pi}_{2\beta-1}(u) =
 \dfrac{ (u^2+iu\sqrt{2}-1)^\beta  - (u^2-iu\sqrt{2}-1)^\beta  }
       {2i\beta\sqrt{2} },
\hat{\pi}_{2\beta}(u) =
 \dfrac{ (u^2+iu\sqrt{2}-1)^\beta  + (u^2-iu\sqrt{2}-1)^\beta  }
       {2 }.
$$
Remark that the polynomial $\hat{\pi}_{2\beta}$ has a meaning despite
its infinite norm, as we only have to check orthogonality with
even powers $u^k$, $k=0,2,\dotsc, 2\beta-2$. Therefore, as
$(1+u^4)^\beta$ is the product of
$ (u^2+iu\sqrt{2}-1)^\beta$  and $ (u^2-iu\sqrt{2}-1)^\beta$, we must
check that the integrals of $u^k/(u^2 \pm iu\sqrt{2} -1)^\beta$ on
$(-\infty,\infty)$ all vanish for $k=0,1,\dotsc, 2\beta-2$. And that
appears by making a contour integral by adding a big semicircle in
the upper or the lower half complex plane, so as to avoid the two
poles (which have the same imaginary part).

We then find $\gamma_{2\beta-1}, \gamma_{\beta-2},\dotsc$ from
the continued fraction expansion about $\infty$
$$
\dfrac{\hat{\pi}_{2\beta}(u)}
      {\hat{\pi}_{2\beta-1}(u)} =
\dfrac{u^{2\beta}-\beta^2 u^{2\beta-2}+\dotsb }
      {u^{2\beta-1}-(\beta^2-1) u^{\beta-3}/3+\dotsb }=
u-\dfrac{ \gamma_{2\beta-1} }
        { u-\dfrac{ \gamma_{2\beta-2} }{u-\dotsb}  }
$$
whence $\gamma_{2\beta-1} = (2\beta^2+1)/3$.
The other $\gamma$'s are more complicated, for instance,
$ \gamma_{2\beta-2}=  2(\beta^2-1)(2\beta^2+7)/[15(2\beta^2+1)]$.
Actually, simpler formulas for $\gamma_n$ are found in the forbidden
region $n> 2\beta-1$\dots
From $\gamma_{2\beta-2},
      \gamma_{2\beta-1}$, and \eqref{ppp} with $k=2\beta-1$, we have
$\rho_{2\beta-1}=-2\beta^2(2\beta^2+1)/3$, whence
$\gamma_{2\beta}=-2\beta^2$.

\item

For a larger half-integer $\alpha$,
  we have to consider orthogonal polynomials
with respect to
$\hat{w}(u)= u^{2\alpha+1}(1+u^4)^{-\beta}$ on $(-\infty,\infty)$, where
$2\alpha+1$ is a positive even number.  The  orthogonal
polynomials $\hat{\pi}_n$ with respect to this weight is a kernel
polynomial built with orthogonal polynomials relative to
the weight
$u^{2\alpha-1}(1+u^4)^{-\beta}$ which is multiplied by $u^2$. The
formula relating the two families of orthogonal polynomials is
$$
 (\hat{\pi}_n(u))_{u^2 \hat{w}(u) } = \dfrac
   { (\hat{\pi}_{n+2}(u))_{\hat{w}(u) }
     -c_n(\hat{\pi}_n(u))_{\hat{w}(u)  }}{u^2},
$$
where $c_n$ is such that the numerator is a multiple of $u^2$.
When $n$ is even, $c_n$ is simply $\hat{u}_{n+2}(0)/
                                    \hat{u}_n(0) = -\gamma_{n+1}$
from the recurrence relation, and
\begin{equation} \label{keven}
 (\hat{\pi}_n(u))_{u^2 \hat{w}(u) } = \dfrac
   { (\hat{\pi}_{n+1}(u))_{\hat{w}(u)}}   {u}\;, \   n \text{\ even}.
\end{equation}
When $n$ is odd, the interesting form is
\begin{equation} \label{kodd}
 (\hat{\pi}_n(u))_{u^2 \hat{w}(u) } = c'_n\dfrac
   { u(\hat{\pi}_{n+3}(u))_{\hat{w}(u) }
    -(u^2-c''_n) (\hat{\pi}_{n+2}(u))_{\hat{w}(u)  }
    }   {u^2}\;, \   n \text{\ odd},
\end{equation}
found from two applications of the recurrence relation. In \eqref{kodd},
$c''_n$ is such that $u^2$ is a factor of the numerator, i.e.,
$c''_n = -\hat{\pi}_{n+3}(0)/
          \hat{\pi}'_{n+2}(0)$, and $c'_n$ is such that the result
is a monic polynomial.


Let us show by induction that two of the orthogonal polynomials with
respect to the weight function $u^{2\alpha+1}(1+u^4)^{-\beta}$
have the following formulas:
\begin{equation} \label{podd}
\hat{\pi}_{2\beta-2\alpha-2}(u) =
\sum_{r=0} (-1)^r \dfrac{ (\beta-\alpha-3/2)
                          (\beta-\alpha-5/2)  \dotsb
                          (\beta-\alpha-1/2-2r)  }
                       {r! (2\alpha+4)(2\alpha+6)\dotsb (2\alpha+2r+2)}
      ( u^2-1)^{\beta-\alpha-3/2-2r}   u^{2r+1}
\end{equation}
\begin{equation} \label{peven}
\hat{\pi}_{2\beta-2\alpha-1}(u) =
\sum_{r=0} (-1)^r \dfrac{ (\beta-\alpha-1/2)
                          (\beta-\alpha-3/2)  \dotsb
                          (\beta-\alpha+1/2-2r)  }
                       {r! (2\alpha+2)(2\alpha+4)\dotsb (2\alpha+2r)}
      ( u^2-1)^{\beta-\alpha-1/2-2r} u^{2r}.
\end{equation}
The formulas are valid when $\alpha=-1/2$, from the expansion by the
binomial theorem of $(u^2 \pm iu\sqrt{2} -1)^\beta$ as
a sum of products of powers of $u^2-1$ and powers of $iu\sqrt{2}$.

Suppose now that \eqref{podd} and \eqref{peven} are true for
$\alpha-1$, so, with $\hat{w}(u)=u^{2\alpha-1}(1+u^4)^{-\beta}$,
$$
(\hat{\pi}_{2\beta-2\alpha}(u))_{\hat{w}} =
\sum_{r=0} (-1)^r \dfrac{ (\beta-\alpha-1/2)
                          (\beta-\alpha-3/2)  \dotsb
                          (\beta-\alpha+1/2-2r)  }
                       {r! (2\alpha+2)(2\alpha+4)\dotsb (2\alpha+2r)}
      ( u^2-1)^{\beta-\alpha-1/2-2r}   u^{2r+1} ,
$$
$$
(\hat{\pi}_{2\beta-2\alpha+1}(u))_{\hat{w}} =
\sum_{r=0} (-1)^r \dfrac{ (\beta-\alpha+1/2)
                          (\beta-\alpha-1/2)  \dotsb
                          (\beta-\alpha+3/2-2r)  }
                       {r! (2\alpha)(2\alpha+2)\dotsb (2\alpha+2r-2)}
      ( u^2-1)^{\beta-\alpha+1/2-2r} u^{2r},
$$
we immediately have \eqref{peven} from \eqref{keven}. For \eqref{podd},
remark first that
$(\hat{\pi}'_{2\beta-2\alpha}(0))_{\hat{w}} =  (-1)^{\beta-\alpha-1/2}$,
$(\hat{\pi}_{2\beta-2\alpha+1}(0))_{\hat{w}} = (-1)^{\beta-\alpha+1/2}$,
so that $c''_n=1$. Then, from \eqref{kodd},
$$
(\hat{\pi}_{2\beta-2\alpha-2}(u))_{u^2\hat{w}(u)} = c'_n u^{-2}\left[
\sum_{r=0} (-1)^r \dfrac{ (\beta-\alpha+1/2)
                          (\beta-\alpha-1/2)  \dotsb
                          (\beta-\alpha+3/2-2r)  }
                       {r! (2\alpha)(2\alpha+2)\dotsb (2\alpha+2r-2)}
      ( u^2-1)^{\beta-\alpha+1/2-2r} u^{2r+1}  \right.
$$
$$
\left. -\sum_{r=0} (-1)^r \dfrac{ (\beta-\alpha-1/2)
                          (\beta-\alpha-3/2)  \dotsb
                          (\beta-\alpha+1/2-2r)  }
                       {r! (2\alpha+2)(2\alpha+4)\dotsb (2\alpha+2r)}
      ( u^2-1)^{\beta-\alpha+1/2-2r}   u^{2r+1} \right],
$$
and \eqref{podd} follows with $\displaystyle{ c'_n =
 -\dfrac{ \beta+\alpha+1/2}{2\alpha(\alpha+1)} }$.

We find now some $\gamma$'s from
$$\dfrac{ \hat{\pi}_{2\beta-2\alpha-1}(u) }
        { \hat{\pi}_{2\beta-2\alpha-2}(u) }   =
$$
$$
 \dfrac{ u^{2\beta-2\alpha-1} -
         \dfrac{\beta^2-(\alpha+1/2)^2 }{2\alpha+2}
         u^{2\beta-2\alpha-3} + \dotsb }
       { u^{2\beta-2\alpha-2} -
         \dfrac{\beta^2-(\alpha+3/2)^2 }{2\alpha+4}
         u^{2\beta-2\alpha-4} + \dotsb }
$$
$$
 =  u -\dfrac{  \gamma_{2\beta-2\alpha-2}  }
             {u - \dfrac{  \gamma_{2\beta-2\alpha-3}  }{u-\dotsb}  }
$$
whence
$\displaystyle{
 \gamma_{2\beta-2\alpha-2} =
         \dfrac{\beta^2-(\alpha+1/2)^2 }{2\alpha+2}
        -\dfrac{\beta^2-(\alpha+3/2)^2 }{2\alpha+4}
}$. The $\gamma$'s with lower subscripts have more complicated
expressions. For instance,
 For higher subscripts, we shall use the
equations \eqref{lorFk} and \eqref{lorrhok} which, after all, were
made for this purpose.
\end{enumerate}

\section{Calculation of nodes and weights.}

Bidiagonal matrix of $\gamma$'s: singular values!
 Danloy \cite[p.~863]{dan1}, Laurie \cite{Laurie}


\section*{Acknowledgments}

V. Pierrard thanks the FNRS for the grant and the Directeur of IASB-BIRA for his
hospitality.



"This paper presents research results of the Belgian Programme on
Interuniversity Attraction Poles, initiated by the Belgian Federal
Science Policy Office.  The scientific responsibility rests with its
author(s)."

\begin{thebibliography}{99}


\bibitem{abra} Abramowitz, M., and I. A. Stegun (Eds.), {\it Handbook of
Mathematical Functions},
Dover, Mincola, N. Y., 1968.


\bibitem{BeR}  S. Belmehdi, A. Ronveaux, Laguerre-Freud's equations
     for  the recurrence coefficients of semi-classical
     orthogonal polynomials,  {\sl J.\ Approx.\ Theory\/}

\bibitem{Bes} D.~Bessis, A new method in the combinatorics of the
    topological expansion, {\sl Comm.\ Math.\ Phys.\/} {\bf 69\/}
    (1979), 147-163.

\bibitem{chi}
   T.S.~Chihara,
   \textsl{An Introduction to Orthogonal Polynomials},
   Gordon and Breach, 1978.

\bibitem{Chin}  R.C.Y.~Chin, A domain decomposition method for generating
            orthogonal polynomials for a Gaussian weight on a finite
            interval, {\sl J. Comp. Phys.\/} {\bf 99\/} (1992) 321-336.

\bibitem{ClS} A.S.\ Clarke, B.\ Shizgal, On the generation of orthogonal
      polynomials using asymptotic methods for recurrence coefficients.
      {\sl J.\ Comp.\ Phys.\ \/} {\bf 104\/} (1993) 140-149.

\bibitem{dan1}B.~Danloy,
   Numerical construction of Gaussian quadrature formulas for $\int
   \sb{0}\sp{1}(-{\rm Log} x)·x\sp{\alpha }·f(x)·dx$ and $\int
   \sb{0}\sp{\infty } E\sb{m}(x)·f(x)·dx$.
  \textsl{Math. Comp.} \textbf{27} (1973) 861--869.  %  863



\bibitem{dan} B.~Danloy, numerous private communications.

\bibitem{Freud}  G.Freud, On the coefficients in the recursion formul\ae\  of
     orthogonal polynomials, {\sl Proc.\ Royal Irish Acad. Sect. A\/}
     {\bf 76\/} (1976), 1-6.


\bibitem{gaut94} Gautschi, W., Algorithm 726: Orthpol - A package of routines
for
generating orthogonal polynomials and Gauss-type quadratures, {\it ACM Trans.
Math. Soft., 20}, 21-62, 1994.



\bibitem{gaut81} Gautschi W., Minimal solutions of three-term recurrence
relations
        and orthogonal polynomials, {\it Math. Comp., vol.~36,
      nr.154}, 547-554, 1981.

\bibitem{gaut04} Gautschi W.,
\textsl{Orthogonal Polynomials - Computation and Approximation },
 Oxford University Press, 2004.

\bibitem{golu} Golub G. H. and J.H. Welsch, Calculation of Gauss quadrature
rules,
   {\it Math. Comp., vol. 23, 106}, 221-230, 1969.

\bibitem{hayd} Haydock R., C.M.M.~Nex, A general terminator for the
    recursion method, {\it J.\ Phys.\ C: Solid State Phys., 18},
  2235-2248, 1985.

\bibitem{kolb}
   D.~Kolb,
   Numerical analysis and orthogonal polynomials,
   \textsl{Appl. Math. Comput.} \textbf{2} (1976), no. 3, 257--272.

\bibitem{Laurie}
   Laurie, Dirk P.,
   Questions related to Gaussian quadrature formulas and two-term
   recursions.
   \textsl{Applications and computation of orthogonal polynomials
    (Oberwolfach, 1998)}, 133--144,
   Internat. Ser. Numer. Math., \textbf{131},
   Birkh\"auser, Basel, 1999.

\bibitem{LeQ} J.S.Lew, D.A.Quarles, Nonnegative solutions of a nonlinear
     recurrence, {\sl J.\ Approx.\ Th.\/} {\bf 38\/} (1983), 357-379.

\bibitem{MagP}
 A.P.~Magnus, Freud's equations for orthogonal polynomials
as discrete Painlev'e
    equations, pp. 228-243 in
  {\sl Symmetries and Integrability of Difference Equations},
        Edited by Peter A. Clarkson \& Frank W. Nijhoff,
   Cambridge U.P., Lond. Math. Soc. Lect. Note Ser. 255, 1999;
 pdf preprint in
\href{http://www.math.ucl.ac.be/~magnus/freud/freudpain.pdf}
  {\texttt{http://www.math.ucl.ac.be/\~{}magnus/freud/freudpain.pdf}}
  (217K).

\bibitem{maks97a}
 Maksimovic, M., V. Pierrard and J. Lemaire, A kinetic model of the solar
wind with Kappa distributions in the corona, {\it Astronomy and
Astrophysics, 324}, 725, 1997a.

\bibitem{maks97b}
 Maksimovic, M., V. Pierrard and P. Riley, Ulysses electron distributions
fitted with Kappa functions: towards a new kinetic model of the solar wind, {\it
Geophys.
Res. Let., 24,} 9, 1151, 1997b.

\bibitem{Nevai} P.~Nevai,

\bibitem{pile96} Pierrard V., and J. Lemaire, Lorentzian ion exosphere model,
{\it
J. Geophys. Res.}, 101, 7923, 1996.


\bibitem{pres}
Press, W. H., S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical
recipes in FORTRAN, Cambridge University Press, 2d Edition, 1992.

\bibitem{shiz81} Shizgal, B., A Gaussian quadrature procedure for use in the
solution of the
Boltzmann equation and related problems, {\it J. Comput. Phys., 41}, 2, pp.
309, 1981.

\bibitem {Sho}  J.A. Shohat, A differential equation for orthogonal
     polynomials, {\sl Duke Math. J.\/} {\bf 5\/} (1939),401-417.

\bibitem{szeg} Szego, G., Orthogonal polynomials,
 American Math. Soc., Providence, 1939.



\end{thebibliography}


\end{document}
