Sorry for perhaps naive questions, I am not at all a specialist in the subject
but I need it for my research.
I know that there are close relations between Riemann-Hilbert problems and
orthogonal polynomials. My precise questions are as follows:


1) There are several well-known methods for the numerical computation of roots of orthogonal polynomials (I am interested in multiprecision here). Can R-H help on this ?


2) A very classical fact on orthogonal polynomials is that there is a continued fraction associated to them. If this CF is written in some sort of canonical form (many exist), I have been told that R-H can give the asymptotics of the coefficients of the CF. How is this done?



Riemann-Hilbert and orthogonal polynomials
Let $P\_{n}(z)=\gamma\_{n}z^{n}+\cdots$ be a sequence of orthonormal polynomials with respect to some weight $w$ on $\mathbb{R}$. Given an $n\geq0$, consider the following Riemann-Hilbert problem for the $2\times2$ matrix-valued function $Y\_{n}(z)$ of the complex variable $z$,


1) $z\to Y\_{n}(z)$ is analytic on $\mathbb{C}\setminus\mathbb{R}$,


2) $Y\_{n,+}(z)=Y\_{n,-}(z)\begin{pmatrix} 1 & w\\0 & 1\end{pmatrix}$, where $Y\_{n,\pm}$ denotes the upper and lower limits of $Y\_{n}$ on $\mathbb{R}$,


3) $Y\_{n}(z)=(I+\mathcal{O}(1/z))\begin{pmatrix} z^{n} & 0 \\ 0 & z^{-n}\end{pmatrix}$ as $z\to\infty$.


The Riemann-Hilbert problem has a unique solution $Y\_{n}$, which is given by
$$Y\_{n}(z)=\begin{pmatrix} P\_{n}(z) & {\displaystyle\dfrac{1}{2i\pi}\int\dfrac{P\_{n}(x)w(x)}{x-z}dx}\\[10pt]
c\_{n-1}P\_{n-1}(z) & {\displaystyle\dfrac{c\_{n-1}}{2i\pi}\int\dfrac{P\_{n-1}(x)w(x)}{x-z}dx}
\end{pmatrix},
\qquad c\_{n-1}=-2i\pi\gamma\_{n-1}^{2},$$
(the jump of $Y\_n$ on the real line comes from the singular integrals on the second row).
Performing a series of transformations on the initial Riemann-Hilbert problem leads to a Riemann-Hilbert problem normalized at infinity, and whose jump matrix is uniformly close to the identity matrix as $n$ tends to infinity. Using the asymptotic expansion in powers of $1/n$ of this jump matrix, one may derive an asymptotic expansion for $Y\_{n}$ and thus for $P\_{n}$. This method is described in
the book ``Orthogonal polynomials and random matrices: a Riemann-Hilbert approach'' by P. Deift.


As is well-known, the polynomials $P\_{n}$ satisfy a three-term recurrence relation
$$
b\_{n-1}P\_{n-1}(z)+(a\_{n}-z)P\_{n}(z)+b\_{n}P\_{n+1}(z)=0,
$$
and asymptotic expansions of the coefficients $a\_{n}$ and $b\_{n-1}$ can also be derived by the above method because
$$
a\_{n}=(Y\_{n,1})\_{11}-(Y\_{n+1,1})\_{11},\qquad b\_{n-1}^{2}=(Y\_{n,1})\_{12}(Y\_{n,1})\_{21},
$$
where
$$
Y\_{n}(z)=(I+z^{-1}Y\_{n,1}+\mathcal{O}(z^{-2}))\begin{pmatrix}
z^{n} & 0\\ 0 & z^{-n}
\end{pmatrix},
$$
see Chapter 3 of Deift's book. Finally, the coefficients in the recurrence relation also appear in the expansion as a continued fraction of
$$
\frac{Q\_{n}(z)}{P\_{n}(z)}=\frac{1}{z-a\_{0}-{\displaystyle\frac{b\_{0}^{2}}{z-a\_{1}-\cdots}}},
$$
where $Q\_{n}$ is the associated polynomial of the second kind. For an application of the method to polynomials orthogonal with respect to the modified Jacobi weight $(1-x)^{\alpha}(1+x)^{\beta}h(x)$ on $[-1,1]$, $h$ a positive real analytic function, see [Arxiv](https://arxiv.org/abs/math/0111252), where, for instance, asymptotic expansions for the recurrence coefficients are explicitly computed (Theorem 1.10).


