International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 49-52, Pages 2761-2772
doi:10.1155/S0161171204401045
Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space
Fred Brackx
, Nele De Schepper
and Frank Sommen
Clifford Research Group, Department of Mathematical Analysis, Ghent University, Galglaan 2, Gent 9000, Belgium
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.