scholarly journals ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS VIA RECURRENCE RELATIONS

2012 ◽  
Vol 10 (02) ◽  
pp. 215-235 ◽  
Author(s):  
X.-S. WANG ◽  
R. WONG
Keyword(s):  
Orthogonal Polynomials ◽  
Legendre Polynomials ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Asymptotic Formulas ◽  
Systematic Technique ◽  
Asymptotics Of Orthogonal Polynomials

We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.

scholarly journals Numerical construction of the generalized Hermite polynomials

2003 ◽  
pp. 49-63
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Stable Method ◽  
Numerical Construction ◽  
Three Term Recurrence Relation

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

scholarly journals Asymptotics for orthogonal polynomials off the circle

2004 ◽  
Vol 2004 (1) ◽  
pp. 37-53
Author(s):  
R. Khaldi ◽  
R. Benzine
Keyword(s):  
Orthogonal Polynomials ◽  
Fixed Points ◽  
Unit Circle ◽  
Positive Measure ◽  
Asymptotic Formulas ◽  
Strong Asymptotics ◽  
Asymptotics Of Orthogonal Polynomials ◽  
The Masses

We study the strong asymptotics of orthogonal polynomials with respect to a measure of the typedμ/2π+∑j=1∞Ajδ(z−zk), whereμis a positive measure on the unit circleΓsatisfying the Szegö condition and{zj}j=1∞are fixed points outsideΓ. The masses{Aj}j=1∞are positive numbers such that∑j=1∞Aj<+∞. Our main result is the explicit strong asymptotic formulas for the corresponding orthogonal polynomials.

scholarly journals Recurrence relations for exceptional Hermite polynomials

2016 ◽  
Vol 204 ◽  
pp. 1-16 ◽  
Author(s):  
D. Gómez-Ullate ◽  
A. Kasman ◽  
A.B.J. Kuijlaars ◽  
R. Milson
Keyword(s):  
Recurrence Relations ◽  
Hermite Polynomials

Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Numerical Methods ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Rational Function ◽  
Recurrence Relations ◽  
Recurrence Coefficients ◽  
Basic Task ◽  
Sobolev Orthogonal Polynomials ◽  
Gauss Type ◽  
Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

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