Recurrence relations for the connection coefficients of classical orthogonal polynomials

2018 ◽  
Vol 29 (7) ◽  
pp. 543-552 ◽  
Author(s):  
Yan-Ping Mu
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relations ◽  
Classical Orthogonal Polynomials ◽  
Connection Coefficients

scholarly journals On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Paul Barry
Keyword(s):  
Orthogonal Polynomials ◽  
Closed Form ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Connection Coefficients ◽  
Riordan Arrays ◽  
Boubaker Polynomials ◽  
Riordan Array

The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

Numerical quadrature

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Recurrence Relations ◽  
Classical Mechanics ◽  
Numerical Quadrature ◽  
Classical Orthogonal Polynomials ◽  
Gaussian Integration ◽  
Physical Problems ◽  
Important Means ◽  
Improper Integrals

This chapter discusses the numerous applications of numerical quadrature (integration) in classical mechanics, in semiclassical approaches to quantum mechanics, and in statistical mechanics; and then describes several ways of implementing integration in C++, for both proper and improper integrals. Various algorithms are described and analyzed, including simple classical quadrature algorithms as well as those enhanced with speedups and convergence tests. Classical orthogonal polynomials, whose properties are reviewed, are the basis of a sophisticated technique known as Gaussian integration. Practical implementations require the roots of these polynomials, so an algorithm for finding them from three-term recurrence relations is presented. On the computational side, the concept of polymorphism is introduced and exploited (prior to the detailed treatment later in the text). The nondimensionalization of physical problems, which is a common and important means of simplifying a problem, is discussed using Compton scattering and the Schrödinger equation as an example.

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