Asymptotics of the Wilson polynomials

Li, Yu-Tian; Wang, Xiangsheng; Wong, R.

doi:10.1142/s0219530519500076

Cited by 7 publications

(5 citation statements)

References 29 publications

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“…in (13)), and its asymptotic behavior near = 1(resp. = −1) in (21), it is readily seen that Re ( ) < 0 (resp. Rẽ( ) < 0) on Σ 1 , Σ 3 (resp.…”

Section: Lemmamentioning

confidence: 99%

“…We should also mention that asymptotic approximations for typical examples of the Freud-like weights have been derived via difference equation methods, including continuous Hahn polynomials, 19 continuous dual Hahn polynomials, 20 and Wilson polynomials. 21 However, in the difference equation approach, the asymptotic formulas are usually valid only on the real axis. On the other hand, the valid region of results presented in this paper will cover the whole complex plane.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

Long

Dai

et al. 2019

Stud Appl Math

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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. K E Y W O R D S asymptotic approximation, asymptotic Freud-like weight, continuous dual Hahn polynomials, Riemann-Hilbert problem ∑ =0 ( ) ( 2 ) ( + ) − ( − + ) − !( − )! , Stud Appl Math. 2020;144:133-163. wileyonlinelibrary.com/journal/sapm

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“…in (13)), and its asymptotic behavior near = 1(resp. = −1) in (21), it is readily seen that Re ( ) < 0 (resp. Rẽ( ) < 0) on Σ 1 , Σ 3 (resp.…”

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

Long

Dai

et al. 2019

Stud Appl Math

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“…However, θ = 2 and one of the transition point being zero is exactly the case of Wilson polynomials. For the results in that case, we refer to [57].…”

Section: 37)mentioning

confidence: 99%

“…The first paper on turning-point problem for difference equation [86] did not appear until 2002; see also [98]. Subsequently, progress seems to have picked up, and results in this area are now nearly as complete as they are for differential equations; see [87], [88], [13] and [57].…”

Section: Introductionmentioning

confidence: 99%

Recent Advances in Asymptotic Analysis

Wong¹,

Zhao²

2022

Preprint

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This is a survey article on an old topic in classical analysis. We present some new developments in asymptotics in the last fifty years. We start with the classical method of Darboux and its generalizations, including an uniformity treatment which has a direct application to the Heisenberg polynomials. We then present the development of an asymptotic theory for difference equations, which is a major advancement since the work of Birkhoff and Trjitzinsky in 1933. A new method was introduced into this field in the nineteen nineties, which is now known as the nonlinear steepest descent method or the Riemann-Hilbert approach. The advantage of this method is that it can be applied to orthogonal polynomials which do not satisfy any differential or difference equations neither do they have any integral representations. As an example, we mention the case of orthogonal polynomials with respect to the Freud weight. Finally, we show how the Wiener-Hopf technique can be used to derive asymptotic expansions for the solutions of an integral equation on a half line.

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“…A lot of investigations have been done under this direction and it turns out to be a great success by the effort of many authors. Among the complete list, we briefly mention [8,11,12,15,16,17,25,35,38,39,40,43,44]. See also an application of this approach to some polynomials from indeterminate moment problems [13] and a survey article on this approach [42].…”

Section: Introductionmentioning

confidence: 99%