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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?

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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, where, for instance, asymptotic expansions for the recurrence coefficients are explicitly computed (Theorem 1.10).

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  • $\begingroup$ Thanks you, this completely answers my second question. Do you have any idea on the first, computing the roots using R-H ? $\endgroup$ – Henri Cohen Apr 16 '19 at 8:35
  • $\begingroup$ Estimates for the polynomials are obtained, in particular, on the support of the measure of orthogonality, so, indeed, one can derive estimates for the zeros as well. Usually the asymptotic estimates in the bulk and at the endpoints of the support are different. I don't know if these estimates are useful for the numerical computation of the zeros. $\endgroup$ – user111 Apr 16 '19 at 19:48

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