The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

Details The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

2007-08-29

arxiv.org/abs/0708.3956v2

Journal of Computational and Applied Mathematics 233 (2009), 775--785

Mathematics

Classical Analysis and ODEs

18 pages, 3 figures, addition of important reference, changes in introduction

Scientific paper

We consider orthogonal polynomials $\{p_{n,N}(x)\}_{n=0}^{\infty}$ on the real line with respect to a weight $w(x)=e^{-NV(x)}$ and in particular the asymptotic behaviour of the coefficients $a_{n,N}$ and $b_{n,N}$ in the three term recurrence $x \pi_{n,N}(x) = \pi_{n+1,N}(x) + b_{n,N} \pi_{n,N}(x) + a_{n,N} \pi_{n-1,N}(x)$. For one-cut regular $V$ we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients $a_{n,n}$ and $b_{n,n}$ have asymptotic expansions as $n \to \infty$ in powers of $1/n^2$ and powers of $1/n$, respectively.

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