Asymptotics of orthogonal polynomials with asymptotic Freud-like weights,Studies in Applied Mathematics

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Asymptotics of orthogonal polynomials with asymptotic Freud-like weights
Studies in Applied Mathematics ( IF 2.7 ) Pub Date : 2019-12-06 , DOI: 10.1111/sapm.12291
Wen‐Gao Long ₁ , Dan Dai ₂ , Yu‐Tian Li ₃ , Xiang‐Sheng Wang ₄

Affiliation

Funding information Research Grants Council, University Grants Committee, Grant/Award Numbers: 11300115, 11303016; Chinese University of Hong Kong, Shenzhen, Grant/Award Number: PF01000861; National Natural Science Foundation of China, Grant/Award Number: 11801480; City University of Hong Kong, Grant/Award Numbers: 7004864, 7005032 Abstract We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann– Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.

更新日期：2019-12-06

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