scholarly journals Orthogonal Polynomials On Ellipses And Their Recurrence Relations

2014 ◽  
Vol 47 (1) ◽  
Author(s):  
Vasile Lauric
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Coordinate Function ◽  
Laurent Polynomials ◽  
Orthogonal Laurent Polynomials ◽  
The Matrix

AbstractIn this note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory

2009 ◽  
Vol 52 (1) ◽  
pp. 95-104 ◽  
Author(s):  
L. Miranian
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Classical Theory ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
The Matrix ◽  
Classical Subject ◽  
First Time

AbstractIn the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

scholarly journals THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

2010 ◽  
Vol 21 (02) ◽  
pp. 145-155 ◽  
Author(s):  
P. ROMÁN ◽  
S. SIMONDI
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Hypergeometric Functions ◽  
Spherical Functions ◽  
Hypergeometric Equation ◽  
Necessary Condition ◽  
Hypergeometric Differential Equation ◽  
The Matrix ◽  
Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

scholarly journals Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

2007 ◽  
Vol 206 (2) ◽  
pp. 950-966 ◽  
Author(s):  
Ruymán Cruz-Barroso ◽  
Leyla Daruis ◽  
Pablo González-Vera ◽  
Olav NjÅstad
Keyword(s):  
Unit Circle ◽  
Quadrature Formulas ◽  
Laurent Polynomials ◽  
Orthogonal Laurent Polynomials
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