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Uniform Asymptotics for Orthogonal Polynomials with Exponential Weights—the Riemann–Hilbert Approach
Author(s) -
Wang Z.,
Wong R.
Publication year - 2005
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2005.01567
Subject(s) - mathematics , monic polynomial , orthogonal polynomials , bounded function , combinatorics , polynomial , exponential function , riemann hypothesis , degree (music) , mathematical analysis , physics , acoustics
Let Q ( x ) = q 2 m x 2 m + q 2 m −1 x 2 m −1 +⋯ be a polynomial of degree 2 m with q 2 m > 0 , and let {π n ( x )} n ≥1 be the sequence of monic polynomials orthogonal with respect to the weight w ( x ) = e − Q ( x ) on . Furthermore, let α n and β n denote the Mhaskar–Rakhmanov–Saff (MRS) numbers associated with Q ( x ). By using the Riemann–Hilbert approach, an asymptotic expansion is constructed for π n ( c n z + d n ) , which holds uniformly for all z bounded away from (−∞, −1) , where and .