§18.25 Wilson Class: Definitions

§18.25(i) Preliminaries

For the Wilson class OP’s with : if the -orthogonality set is , then the role of the differentiation operator in the Jacobi, Laguerre, and Hermite cases is played by the operator followed by division by , or by the operator followed by division by . Alternatively if the -orthogonality interval is , then the role of is played by the operator followed by division by .

Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials , continuous dual Hahn polynomials , Racah polynomials , and dual Hahn polynomials .

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

Orthogonality

range for

Constraints

;

nonreal parameters in conjugate pairs

;

nonreal parameters in conjugate pairs

or or

for further constraints see (18.25.1)

or

¶ Further Constraints for Racah Polynomials

If , then the weights will be positive iff one of the following eight sets of inequalities holds:

The first four sets imply , and the last four imply .