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.

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 Parallel Software to Offset the Cost of Higher Precision

2021 ◽  
Vol 40 (2) ◽  
pp. 59-64
Author(s):  
Jan Verschelde
Keyword(s):  
Power Series ◽  
Parallel Algorithms ◽  
Series Expansions ◽  
Use Case ◽  
Double Precision ◽  
Algebraic Space ◽  
Space Curves ◽  
Computational Overhead ◽  
Power Series Expansions ◽  
The Cost

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

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