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Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

Percy Deift, Mateusz Piorkowski

TL;DR

This paper determines the large-$n$ asymptotics of the recurrence coefficients for orthogonal polynomials on $(-1,1)$ with the logarithmic weight $w(x)=\log\left(\frac{2}{1-x}\right)$. Using the nonlinear steepest descent analysis of the Fokas--Its--Kitaev RH problem and a carefully designed comparison to Legendre-type problems, the authors overcome the challenge posed by the edge singularity at $x=+1$ and the zero at $x=-1$ by introducing ${}^\star$RH problems and a Legendre resolvent. The main results are the explicit asymptotics $a_n=\frac{1}{4n^2}-\frac{3}{16n^2\log^2 n}+O(\frac{1}{n^2\log^3 n})$ and $b_n=\frac{1}{2}-\frac{1}{16n^2}-\frac{3}{32n^2\log^2 n}+O(\frac{1}{n^2\log^3 n})$, confirming a special Magnus conjecture case for this weight and extending earlier CD findings to the boundary case $k=1$ with a zero at $-1$. The work highlights a novel mechanism for uniform invertibility via Legendre resolvents and local parametrices, enabling precise control of the logarithmic singularity’s impact on the RH analysis.

Abstract

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by Magnus and extends earlier results by Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at $x = +1$.

Recurrence Coefficients for Orthogonal Polynomials with a Logarithmic Weight Function

TL;DR

This paper determines the large- asymptotics of the recurrence coefficients for orthogonal polynomials on with the logarithmic weight . Using the nonlinear steepest descent analysis of the Fokas--Its--Kitaev RH problem and a carefully designed comparison to Legendre-type problems, the authors overcome the challenge posed by the edge singularity at and the zero at by introducing RH problems and a Legendre resolvent. The main results are the explicit asymptotics and , confirming a special Magnus conjecture case for this weight and extending earlier CD findings to the boundary case with a zero at . The work highlights a novel mechanism for uniform invertibility via Legendre resolvents and local parametrices, enabling precise control of the logarithmic singularity’s impact on the RH analysis.

Abstract

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure on . The asymptotic formula confirms a special case of a conjecture by Magnus and extends earlier results by Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at .
Paper Structure (23 sections, 32 theorems, 111 equations, 7 figures, 1 table)

This paper contains 23 sections, 32 theorems, 111 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. State of the art
  4. Relation of the present work to CD
  5. Outline of the paper
  6. Notation
  7. Auxiliary functions used in the RH analysis
  8. The model and Legendre weight functions
  9. The Szegő functions
  10. The Riemann--Hilbert problem for orthogonal polynomials
  11. Fokas--Its--Kitaev RH problem for the logarithmic weight
  12. Normalized Fokas--Its--Kitaev RH problem for the logarithmic weight
  13. Logarithmic RH problem
  14. Model RH problem
  15. Legendre RH problem
  16. ...and 8 more sections

Key Result

Theorem 1.1

The recurrence coefficients $\lbrace a_n\rbrace_{n=0}^\infty$, $\lbrace b_n \rbrace_{n=0}^\infty$ of the orthogonal polynomials with orthogonality measure $w(x){\rm d}x$, with $w$ given in wlog, satisfy and

Figures (7)

  • Figure 1: Plot of the weight function $w(x)$.
  • Figure 2: A contour with a point of intersection.
  • Figure 3: The contour $\gamma$ encircling the point $z$.
  • Figure 4: Lens-shaped jump contour $\Sigma$.
  • Figure 5: The change of variables $z \to \xi$.
  • ...and 2 more figures

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: CD
  • Remark 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Corollary 2.4: CD
  • Corollary 2.5
  • Remark 2.6
  • ...and 27 more