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{
\baselineskip=14pt
\noindent
\centerline{\fsmall Stieltjes meeting, Delft, 31 Oct.-4 Nov. 1994}
\centerline{\fgrand Problem}
\centerline{\fsmall
 for Mathematicaists, Reducectionists, Macsymalists, and other Mapleators}
\centerline
{\fgrand Painlev\'e equations for semi-classical recurrence coefficients.}
}

\vskip  2mm

\centerline{Alphonse P. Magnus}

\centerline
{\fsmall Institut Math\'ematique,  Universit\'e Catholique de Louvain,
 Chemin du Cyclotron 2,
 B-1348 Louvain-la-Neuve,
 Belgium.}

\centerline{\fsmall e-mail:  {\ttt magnus@anma.ucl.ac.be}}
\centerline{\fsmall www:  {\ttt http://www.math.ucl.ac.be/\~{}magnus}}

  A shorter version is in {\sl Constructive Approximation}, {\bf 12}
 (1996) 303--306.


\bigskip

}  % fin page titre

\displayindent=5pt

{\bf Background:\/}
orthonormal polynomials $p_n(x)=\gamma_n x^n+\cdots\ \ $:
$$ a_{n+1}p_{n+1}(x)=(x-b_n)p_n(x)-a_n p_{n-1}(x) \eqno(1)$$
related to a semi-classical weight\ $w:  \qquad \qquad
   \dfrac{w'(x)}{w(x)}=\dfrac{2V(x)}{W(x)}$  \hfill(2)\break
where $V$ and $W$ are polynomials, satisfy differential relations
$$Wp_n' = (\Omega_n-V)p_n -a_n \Theta_n p_{n-1} \eqno(3)$$
where $\Omega_n$ and $\Theta_n$ are polynomials of degrees $\le m-1$
and $m-2$ (if degrees $V$ and $W \le m-1$ and $m$), with
\qquad
$ \Omega_{n+1}(x)=(x-b_n)\Theta_n(x)-\Omega_n(x), \hfill (4)$
$$ (x-b_n)(\Omega_{n+1}(x)-\Omega_n(x))=W(x)+a_{n+1}^2\Theta_{n+1}(x)-
                                             a_n^2\Theta_n(x)$$
representing nonlinear recurrence relations for the $a_n$'s and
$b_n$'s [Laguerre, $19^{\rm th}$ century].  See for instance [2,7,12].

{\bf Statement:\/} {\sl if, in addition to $(2)$, the weight $w$
depends on a parameter $t$ so that
$$ \dfrac{\partial w(x,t)/\partial t}{w(x,t)} =
 \dfrac{T(x,t)}{W(x,t)}, \eqno(5)$$
where $T$ and $W$ are polynomials in $x$ and $t$ $($ the same $W$,
which will now be written $W(x,t))$, each recurrence coefficient
$a_n^2(t)$ and $b_n(t)$ satisfy a nonlinear differential equation having
Painlev\'e property $($movable singular points are only poles$)$ .
} [many people, $20^{\rm th}$ century].  See, for instance,
[1,3 (pp.~399--402), 4,5,6,8,11 (bottom of p.~363)].

I tried in [9] to show this statement for generalized Jacobi weights.

I have an (incomplete) elementary proof of this for {\bf generalized
Jacobi weights\/}
$$\left.
  \eqalign{
     w(x,t) &= 0, \qquad x<x_1(t) {\rm\ or\ } x>x_m(t) \cr
  &= C_j \prod_{k=1}^m |x-x_k(t)|^{\alpha_k},
  \quad x_j(t)<x<x_{j+1}(t), \;\;j=1,\ldots,m-1.\cr
          }   \right\}\eqno(6)
$$
where therefore $W(x,t)=\prod_1^m(x-x_k(t))$ and (2) and (5)
become
$$ \dfrac{w'(x,t)}{w(x,t)}=\sum_{k=1}^m \dfrac{\alpha_k}
                                            {x-x_k(t)}
  {\rm\ and\ }
   \dfrac{\partial w(x,t)/\partial t}{w(x,t)}=
  -\sum_{k=1}^m \dfrac{\alpha_k \dot x_k(t)} {x-x_k(t)}
\eqno(7)
$$
(N.B. so, the $C_j$'s and the $\alpha_j$'s do {\it not\/} depend
on $t$; remark also that $\alpha_k=2V(x_k)/W'(x_k)$,
$\Omega_n(x)=(n+(\sum_1^m \alpha_k)/2)x^{m-1}+\cdots,\
\Theta_n(x)=(2n+1+\sum_1^m\alpha_k)x^{m-2}+\cdots $).
A differential system involving $a_n$ and $b_n$ is then
$$ \dfrac{\dot a_n}{a_n} = \dfrac12 \sum_{k=1}^m
    \dfrac{(\Theta_n(x_k)-\Theta_{n-1}(x_k))\dot x_k}{W'(x_k)},
  \eqno(8) $$
$$ \dot b_n       =          \sum_{k=1}^m
    \dfrac{(\Omega_{n+1}(x_k)-\Omega_n(x_k))\dot x_k}{W'(x_k)}
         = \sum_{k=1}^m
    \dfrac{((x_k-b_n)\Theta_n(x_k)-2\Omega_n(x_k))\dot x_k}{W'(x_k)},
  \eqno(9) $$
$$ \dfrac{\dot\gamma_n}{\gamma_n}=-\dfrac12
  \sum_{k=1}^m \dot x_k \Theta_n(x_k)/W'(x_k)
 \eqno(10) $$
$$ \dfrac d{d t} \dfrac{\Theta_n(x_j)}{W'(x_j)} =
 2\dfrac{\dot\gamma_n}{\gamma_n}  \dfrac{\Theta_n(x_j)}{W'(x_j)} -2
\sum_{k\ne j} \dfrac{\dot x_j -\dot x_k}{x_j-x_k}
    \dfrac{\Theta_n(x_k)\Omega_n(x_j)-\Theta_n(x_j)\Omega_n(x_k)}
      {W'(x_j)W'(x_k)},\eqno(11)$$
$$\hskip-12pt
   \dfrac d{d t} \dfrac{\Theta_{n-1}(x_j)}{W'(x_j)} =
 -2\dfrac{\dot\gamma_{n-1}}{\gamma_{n-1}}  \dfrac{\Theta_{n-1}(x_j)}{W'(x_j)}
  +2
\sum_{k\ne j} \dfrac{\dot x_j -\dot x_k}{x_j-x_k}
 \dfrac{\Theta_{n-1}(x_k)\Omega_n(x_j)-\Theta_{n-1}(x_j)\Omega_n(x_k)}
      {W'(x_j)W'(x_k)}, \eqno(12)$$
$$\dfrac d{d t} \dfrac{\Omega_n(x_j)}{W'(x_j)} =
a_n^2 \sum_{k\ne j} \dfrac{\dot x_j -\dot x_k}{x_j-x_k}
 \dfrac{\Theta_n(x_j)\Theta_{n-1}(x_k)-\Theta_n(x_k)\Theta_{n-1}(x_j)}
      {W'(x_j)W'(x_k)},\eqno(13)$$

{\fgrand Problem:\/} eliminate the $\Theta$'s and the $\Omega$'s in
$(8-13)$ so to exhibit the scalar differential equations for $a_n(t)$ and
$b_n(t)$, at least in the $m=3$ case ($W$ of degree 3 in $x$ and 1 in
$t$), and in the even $m=5$ case ($w(x,t)=|x|^\alpha|x^2-x_1^2|^\beta
|x^2-x_2^2|^\gamma, b_n=0$), see [8] for confluent cases and [9] for the
even case.

{\bf Proof of } $(8-14)$

{\eightpoint

The proof is valid only when $\alpha_k >0$, so that
$$ q_n(x)=\int_{\R} \dfrac{w(u)p_n(u)}{x-u} du \eqno(14)$$
is finite at the $x_k$'s. Then, $\dfrac{d}{dt}\int_\R w(u,t)p_n(u,t)
p_{n-i}(u,t)du=0,\quad i=0,1,\ldots$:
$$\sum_{k=1}^m \alpha_k \dot x_k q_n(x_k)p_{n-i}(x_k)
+\int_\R w(u,t){\dfrac{\partial p_n(u,t)}{\partial t}}
p_{n-i}(u,t)du +{\dfrac{\dot\gamma_n}{\gamma_n}}\delta_{i,0}=0,$$
yielding the coefficients of the orthogonal expansion of
$$\displaylines{
 \hskip-12pt
 \dfrac{\partial p_n(x,t)}{\partial t} =
  -\dfrac{\dot\gamma_n}{\gamma_n} p_n(x,t)
  -\sum_{k=1}^m\alpha_k \dot x_k q_n(x_k,t)
  \sum_{i=0}^n p_{n-i}(x_k,t)p_{n-i}(x,t) \hfill(15)\cr
 \hskip 13pt =-\dfrac{\dot\gamma_n}{\gamma_n} p_n(x,t)
  -\sum_{k=1}^m\alpha_k \dot x_k q_n(x_k,t)
 \left[a_n(t)
 \dfrac{p_n(x,t)p_{n-1}(x_k,t)-p_n(x_k,t)p_{n-1}(x,t)}
       {x-x_k} + p_n(x_k,t)p_n(x,t)
    \right] \hfill(16)\cr}      $$
(Christoffel-Darboux). From the coefficient of $p_n$:
$$ \dfrac{\dot\gamma_n}{\gamma_n}=-\dfrac12
  \sum_{k=1}^m \alpha_k \dot x_k q_n(x_k)p_n(x_k)
$$
Now, from (14), (2), (3), (4), and (1)
$$Wq'_n=(\Omega_n+V)q_n-a_n\Theta_n q_{n-1},
Wp'_{n-1}=a_n\Theta_{n-1} p_n -(\Omega_n+V)p_{n-1},
Wq'_{n-1}=a_n\Theta_{n-1} q_n -(\Omega_n-V)q_{n-1},
$$
so that, using $p_nq_{n-1}-p_{n-1}q_n=1/a_n$,
$$\Theta_n(x_k)=2V(x_k)p_n(x_k)q_n(x_k)=\alpha_k W'(x_k)p_n(x_k)q_n(x_k),
\Omega_n(x_k)-V(x_k)=a_n\alpha_k W'(x_k)q_n(x_k)p_{n-1}(x_k),  \eqno(17)
$$
giving (10),
(8) follows from $\gamma_{n-1}=a_n\gamma_n$: $\dot a_n/a_n=
            \dot\gamma_{n-1}/\gamma_{n-1}-\dot\gamma_n/\gamma_n$.
Now, (16) becomes
$$ \dfrac{\partial p_n(x,t)}{\partial t} =
   \dfrac{\dot\gamma_n}{\gamma_n} p_n(x,t)-
 {\displaystyle \sum_{k=1}^m \dot x_k\;\dfrac{(\Omega_n(x_k)-V(x_k))p_n(x)
      -a_n\Theta_n(x_k)p_{n-1}(x)}{W'(x_k)(x-x_k)}}.\eqno(18)$$
For (9), consider
$$\dfrac{\partial p_n(x,t)}{\partial t} =
  \dfrac{\partial [ \gamma_n(t)x^n-\gamma_n(t)(b_0(t)+\cdots
                   +b_{n-1}(t))x^{n-1}+\cdots ]}{\partial t} =
 \dfrac{\dot\gamma_n}{\gamma_n}p_n -\dfrac{\dot b_0+\cdots
        +\dot b_{n-1}}{a_n} p_{n-1}+\cdots $$
and the coefficient of $p_{n-1}$ in (15):
$$-\dfrac{\dot b_0+\cdots+b_{n-1}}{a_n} =
  -\sum_1^m \alpha_k \dot x_k q_n(x_k)p_{n-1}(x_k) =
 -\dfrac1{a_n} \sum_1^m \dot x_k \dfrac{\Omega_n(x_k)-V(x_k)}{W'(x_k)}\;.
$$

For (11-13), we need the $t-$derivatives of $p_n(x_j(t),t)$,
                                       $q_n(x_j(t),t)$, etc.

$dp_n(x_j(t),t)/dt = (\partial p_n(x,t)/\partial t)(x_j) +
                 \dot x_j p_n'(x_j)$

$(3)\Rightarrow\;
 p_n'(x)=\dfrac{\Omega_n(x)-V(x)}{W(x)}p_n(x)-a_n\dfrac{\Theta_n(x)}{W(x)}
  p_{n-1}(x)=
 {\displaystyle \sum_{k=1}^m \dfrac{(\Omega_n(x_k)-V(x_k))p_n(x)
      -a_n\Theta_n(x_k)p_{n-1}(x)}{W'(x_k)(x-x_k)}}$,
and use (18):
$$dp_n(x_j(t),t)/dt =
   \dfrac{\dot\gamma_n}{\gamma_n} p_n(x_j,t) +
     \sum_{\scriptstyle k=1\atop \scriptstyle k\ne j}^m
      (\dot x_j-\dot x_k)\;\dfrac{(\Omega_n(x_k)-V(x_k))p_n(x_j)
      -a_n\Theta_n(x_k)p_{n-1}(x_j)}{W'(x_k)(x_j-x_k)}
    ,\eqno(19)$$

$$\eqalign{ \hskip -52pt
  \dfrac{\partial q_n(x,t)}{\partial t} &=
  \dfrac{\partial \int_\R \dfrac{w(u,t)p_n(u,t)}{x-u}du}{\partial t} \cr
 = - &\sum_1^m \alpha_k \dot x_k \int_\R \dfrac{w(u)p_n(u)}{(u-x_k)(x-u)}
   du + \int_\R w(u)\dfrac{\dot\gamma_n p_n(u)}{\gamma_n (x-u)}du
   -\sum_1^m  \dot x_k
  \int_\R w(u)\dfrac{[\Omega_n(x_k)-V(x_k)]p_n(u)-a_n\Theta_n(x_k)p_{n-1}(u)}
                    {W'(x_k)(u-x_k)(x-u)}
   du \cr
 &= \dfrac{\dot\gamma_n}{\gamma_n}q_n(x) -
    \sum_1^m  \dot x_k
  \dfrac{[\Omega_n(x_k)+V(x_k)](q_n(x)-q_n(x_k))
  -a_n\Theta_n(x_k)(q_{n-1}(x)-q_{n-1}(x_k))}{W'(x_k)(x-x_k)}\cr
   }
$$
$$dq_n(x_j(t),t)/dt =
   \dfrac{\dot\gamma_n}{\gamma_n} q_n(x_j,t) +
     \sum_{\scriptstyle k=1\atop \scriptstyle k\ne j}^m
      (\dot x_j-\dot x_k)\;\dfrac{(\Omega_n(x_k)+V(x_k))q_n(x_j)
      -a_n\Theta_n(x_k)q_{n-1}(x_j)}{W'(x_k)(x_j-x_k)}
    ,$$
and we get (11) as $\alpha_j d(p_n(x_j)q_n(x_j))/dt$ (from (17)).
For (12): use (11) with $n-1$ instead of $n$, use $\Omega_{n-1}(x)=
(x-b_{n-1}\Theta_{n-1}(x)-\Omega_n(x)$ from (4), and (12) follows from
(10) and also $\sum_1^m \Theta_{n-1}(x_k)/W'(x_k)=$ coefficient of
$x^{m-1}$ of $\Theta_{n-1}=0$). Finally, we need
$d p_{n-1}(x_j(t),t)/dt$ for (13): take (19) with $n-1$, replace
$-a_{n-1}p_{n-2}(x_j)$ by $a_np_n(x_j)-(x_j-b_{n-1})p_{n-1}(x_j)$:
$$dp_{n-1}(x_j(t),t)/dt =
   -\dfrac{\dot\gamma_{n-1}}{\gamma_{n-1}} p_{n-1}(x_j,t) -
     \sum_{\scriptstyle k=1\atop \scriptstyle k\ne j}^m
      (\dot x_j-\dot x_k)\;\dfrac{(\Omega_n(x_k)+V(x_k))p_{n-1}(x_j)
      -a_n\Theta_{n-1}(x_k)p_n(x_j)}{W'(x_k)(x_j-x_k)}
    .$$


} %fin eightpoint

{\fgrand Exercise:\/} {\sl show that the moment
$\mu_n=\int_\R w(u) (u-x_1)^n du$ of $(6)$ enters the solution of the
following differential
system as} $\mu_n=\nu_{n,1}$,   \hfill\break
{\sl where\/} $\nu_{n,j}=\int_\R w(u)\prod_1^m(u-x_k)^{\beta_k}
     (u-x_j)^{-1}du$:

$$ \dot\nu_{n,j} = \sum_{\scriptstyle k=1\atop \scriptstyle k\ne j}^m
       (\dot x_j-\dot x_k)(\alpha_k+\beta_k)
       \dfrac{\nu_{n,j}-\nu_{n,k}}{x_j-x_k}\,,\qquad
       j=1,\ldots,m $$
{\sl with\/} $\beta_1=n+1,\beta_2=\ldots=\beta_m=1.$

This shows that the moments of these generalized Jacobi weights are
regular functions of $t$ in any region where the $x_k(t)$'s are
distinct. As $a_n^2(t)$ and $b_n(t)$ are ratios of determinants of
moments, the movable singular points of their equations can only be
poles (the zeros of the denominator determinants).


{\fgrand References.}\smallskip \frenchspacing

\item{[1]} A.I.~Aptekarev, A.~Branquinho, F.~Marcell\'an, 
     Toda-type differential equations for the recurrence coefficients of
    orthogonal polynomials and Freud transformation,
  {\sl Pr\'e-Publica\c coes Univ. Coimbra, Dep. Matem\'atica} 96--04 (Feb.~1996).

 
\item{[2]}  S. BELMEHDI, A. RONVEAUX, About non linear systems
     satisfied by the recurrence coefficients of semiclassical
     orthogonal polynomials,  {\sl J.\ Approx.\ Th.\/}
     {\bf 76\/} (1994), 351-368.
 
 
\item{[3]} D.V.~CHUDNOVSKY, Riemann monodromy problem, isomonodromy
      deformation eqations and completely integrable systems, pp.385-447
      {\it in\/} {\sl Bifurcation Phenomena in Mathematical Physics and
      Related Topics, Proceedings  Carg\`ese, 1979\/} (C.BARDOS \&
      D.BESSIS, editors), NATO ASI series~C, vol. {\bf 54\/}, D.Reidel,
      Dordrecht, 1980.

\item{[4]}        G.V.~CHUDNOVSKY, Pad\'e approximation and the
      Riemann monodromy problem, {\it ibidem\/}, pp.449-510.
 
\item{[5]} D.V.~Chudnovsky, G.V.~Chudnovsky, Explicit continued fractions
     and quantum gravity, {\sl Acta Applicandae Math.} {\bf 36} (1994)
     167--185.

 
\item{[6]} A.S.\ FOKAS, A.R.\ ITS, A.V.\ KITAEV, Discrete
        Painlev\'e equations and their appearance in quantum gravity,
       {\sl Commun.\ Math.\ Phys.\ \/} {\bf 142\/} (1991) 313-344;
        The isomonodromy
   approach to matrix models in 2D quantum gravity,
   {\it ibidem\/} {\bf 147\/} (1992) 395-430.
 
\item{[7]}  E.~LAGUERRE, Sur la r\'eduction en fractions continues d'une
      fraction qui satisfait \`a une \'equation diff\'erentielle lin\'eaire
      du premier ordre dont les coefficients sont rationnels,
      {\sl J. Math. Pures Appl. (4)\/} {\bf 1\/} (1885), 135-165 =
      pp. 685-711 {\it in\/}
      {\sl Oeuvres\/}, Vol.II, Chelsea, New-York 1972.
 
\item{[8]} A.P.~MAGNUS, Painlev\'e-type differential equations for
        the recurrence  coefficients of semi-classical orthogonal
         polynomials,  {\sl J.\ Comp.\ Appl.\ Math.} {\bf 57} (1995) 215--237.
       Preliminary preprint:
 {\tt ftp://unvie6.un.or.at/siam/opsf/pailevemagnus.tex}
 
 
\item{[9]} A.P.~MAGNUS,
 Asymptotics for the simplest generalized Jacobi polynomials recurrence
  coefficients from Freud's equations: numerical explorations, 
{\sl Annals of Numerical Mathematics} {\bf  2} (1995) 311--325.
 Preprint in  directory
 {\tt ftp://unvie6.un.or.at/siam/opsf/magnus/}
 
\item{[10]} A.P.~MAGNUS,
 Problem: Painlev\'e equations for semi-classical recurrence coefficients,
 preprint
 {\tt ftp://unvie6.un.or.at/siam/opsf/magnus\_painleve\_problem.tex}
 
 
\item{[11]} J.NUTTALL, Asymptotics of diagonal Hermite-Pad\'e polynomials,
 {\sl J.Approx.\ Th.\ } {\bf 42\ } (1984) 299-386.
 
\item {[12]}  J.A. SHOHAT, A differential equation for orthogonal
     polynomials, {\sl Duke Math. J.\/} {\bf 5\/} (1939),401-417.
 

   {\bf  Acknowledgements.} W. Al-Salam (1926--1996), Professor of Mathematics at
     the University of Alberta, organized an electronic repository of preprints
    about orthogonal polynomials and special functions. He always accepted
    very kindly the burden of this most useful service. The ftp site is now
    maintained by H.~Haubold at {\tt ftp://unvie6.un.or.at/siam/opsf/}.

\bye
