On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials

TL;DR

This paper proves Magnus's conjecture on the asymptotic behavior of recurrence coefficients for a broad class of orthogonal polynomials with singular weights, using advanced Riemann-Hilbert analysis.

Contribution

It rigorously confirms Magnus's conjecture for generalized weights with analytic factors, extending previous results to more complex weight functions.

Findings

Magnus's conjecture is validated for generalized weights.

Asymptotic formulas involve confluent hypergeometric functions.

Methodology employs steepest descent on Riemann-Hilbert problems.

Abstract

In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$ (1-x)^\alpha (1+x)^\beta |x_0 - x|^\gamma \times a jump at x_0, $$ with $\alpha, \beta, \gamma>-1$ and $x_0 \in (-1,1)$. We show rigorously that Magnus' conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of [-1,1] and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at $x_0$ has to be carried out in terms of confluent hypergeometric functions.