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

✍ Scribed by A.B.J. Kuijlaars; P.M.J. Tibboel

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 661 KB
• Volume: 233
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2009.02.090

✦ Synopsis

We consider orthogonal polynomials {p n,N (x)} ∞ n=0 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

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 → ∞ in powers of 1/n 2 and powers of 1/n, respectively.