In this chapter we study orthogonal polynomials that generalize the classical orthogonal polynomials in the sense that they satisfy a second order differential–difference equation. They are orthogonal with respect to weight functions that contain a number of parameters, and they reduce to the classical orthogonal polynomials when some of the parameters go to zero. Explicit formulae for reproducing kernels are derived for some families of orthogonal polynomials.

Generalized Classical Orthogonal Polynomials on the Ball

The classical orthogonal polynomials on the ball Bd are discussed in Subsection 2.3.2.

Definition and differential–difference equations

Definition 6.1.1 Let hK be a reflection invariant weight function as in Definition 5.1.1. Define a weight function on Bd by

The polynomials that are orthogonal to W∧ are called generalized classical orthogonal polynomials on Bd.

Recall that c′h is the normalization constant for h2 K over S∧1; it follows from the polar coordinates that the normalization constant of W∧ is given by

where crd-i is the surface area of Sd~1, jK is the homogeneous degree of hK. Again we shall write 7 for 7∧ in the proofs. If K - 0 or W being the orthogonal group, then hK(x) = 1 and the weight function W∧(x) is reduced to W∧(x) = (1 - \\x\\2Y-∧2, the weight function of the classical orthogonal polynomials on the ball.

We study these polynomials by relating them to /i-harmonics. Recall Theorem 3.8.4 which gives a correspondence between orthogonal polynomials on Bd and those on Sd.