Askey scheme

In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , and has since been extended by and to cover basic orthogonal polynomials.
Askey scheme for hypergeometric orthogonal polynomials.
 give the following version of the Askey scheme:
Askey scheme for basic hypergeometric orthogonal polynomials.
 give the following scheme for basic hypergeometric orthogonal polynomials:
Completeness.
While there are several approaches to constructing still more general families of orthogonal polynomials, it is usually not possible to extend the Askey scheme by reusing hypergeometric functions of the same form. For instance, one might naively hope to find new examples given by
above formula_8 which corresponds to the Wilson polynomials. This was ruled out in under the assumption that the formula_9 are degree 1 polynomials such that
for some polynomial formula_11.