Asymptotics of the Wilson polynomials

2019 ◽  
Vol 18 (02) ◽  
pp. 237-270 ◽  
Author(s):  
Yu-Tian Li ◽  
Xiang-Sheng Wang ◽  
Roderick Wong
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Asymptotic Expansions ◽  
Transition Point ◽  
Linear Difference Equations ◽  
Variable Formula ◽  
Free Zone ◽  
Wilson Polynomials

In this paper, we study the asymptotic behavior of the Wilson polynomials [Formula: see text] as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable [Formula: see text] is fixed and (ii) when the variable is rescaled as [Formula: see text] with [Formula: see text]. Case (ii) has two subcases, namely, (a) zero-free zone ([Formula: see text]) and (b) oscillatory region [Formula: see text]. Corresponding results are also obtained in these cases (iii) when [Formula: see text] lies in a neighborhood of the transition point [Formula: see text], and (iv) when [Formula: see text] is in the neighborhood of the transition point [Formula: see text]. The expansions in the last two cases hold uniformly in [Formula: see text]. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations.

scholarly journals Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis

2020 ◽  
Vol 10 (02) ◽  
pp. 2050003
Author(s):  
Diego Dominici
Keyword(s):  
Power Series ◽  
Asymptotic Analysis ◽  
Orthogonal Polynomials ◽  
Asymptotic Expansions ◽  
Linear Functional ◽  
Series Expansions ◽  
Recurrence Coefficients ◽  
Variable Formula ◽  
Polynomial Sequences ◽  
Power Series Expansions

We study the three-term recurrence coefficients [Formula: see text] of polynomial sequences orthogonal with respect to a perturbed linear functional depending on a variable [Formula: see text] We obtain power series expansions in [Formula: see text] and asymptotic expansions as [Formula: see text] We use our results to settle some conjectures proposed by Walter Van Assche and collaborators.

scholarly journals LINEAR DIFFERENCE EQUATIONS WITH A TRANSITION POINT AT THE ORIGIN

2013 ◽  
Vol 12 (01) ◽  
pp. 75-106 ◽  
Author(s):  
LI-HUA CAO ◽  
YU-TIAN LI
Keyword(s):  
Difference Equation ◽  
Asymptotic Expansions ◽  
Transition Point ◽  
Bessel Functions ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Small Shift ◽  
Linearly Independent ◽  
Uniform Asymptotic Expansion ◽  
Uniform Asymptotic Expansions

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation [Formula: see text] where An and Bn have asymptotic expansions of the form [Formula: see text] with θ ≠ 0 and α0 ≠ 0 being real numbers, and β0 = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t1 = 0, where t = (n + τ0)-θx and τ0 is a small shift. In particular, it is shown how the Bessel functions Jν and Yν get involved in the uniform asymptotic expansions of the solutions to the above linear difference equation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight xα exp (-qmxm), x > 0, where m is a positive integer, α > -1 and qm > 0.

Asymptotic behavior of a system of higher order linear difference equations

2001 ◽  
Vol 47 (7) ◽  
pp. 4667-4677 ◽  
Author(s):  
Hideaki Matsunaga ◽  
Ryuzou Ogita ◽  
Kouichi Murakami
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Higher Order ◽  
Linear Difference Equations ◽  
Order Linear

scholarly journals INVERSE FACTORIAL-SERIES SOLUTIONS OF DIFFERENCE EQUATIONS

2004 ◽  
Vol 47 (2) ◽  
pp. 421-448 ◽  
Author(s):  
A. B. Olde Daalhuis
Keyword(s):  
Hypergeometric Function ◽  
Difference Equations ◽  
Asymptotic Expansions ◽  
Subject Classification ◽  
Series Solutions ◽  
Linear Difference Equations ◽  
Secondary 33C05 ◽  
Rank One ◽  
Factorial Series ◽  
Primary 34E05

AbstractWe obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large $a$ asymptotics of the hypergeometric function ${}_2F_1(a,b;c;x)$.AMS 2000 Mathematics subject classification: Primary 34E05; 39A11. Secondary 33C05

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