International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 52, Pages 2761-2772
doi:10.1155/S0161171204401045
Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space
Clifford Research Group, Department of Mathematical Analysis, Ghent University, Galglaan 2, Gent 9000, Belgium
Received 8 January 2004
Copyright © 2004 Fred Brackx et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into an orthogonality relation on the real axis by means of the so-called Clifford-Heaviside functions. Consequently, appropriate orthogonal polynomials on the real axis give rise to Clifford algebra-valued orthogonal polynomials in the unit ball. Three specific examples of such orthogonal polynomials in the unit ball are discussed, namely, the generalized Clifford-Jacobi polynomials, the generalized Clifford-Gegenbauer polynomials, and the shifted Clifford-Jacobi polynomials.