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 Koekoek, Lesky & Swarttouw to cover basic orthogonal polynomials.
Askey scheme for hypergeometric orthogonal polynomials
Koekoek, Lesky & Swarttouw give the following version of the Askey scheme:;: Wilson | Racah
;: Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn
;: Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk
;: Laguerre | Bessel | Charlier
;: Hermite
Here indicates a hypergeometric series representation with parameters
Askey scheme for basic hypergeometric orthogonal polynomials
Koekoek, Lesky & Swarttouw give the following scheme for basic hypergeometric orthogonal polynomials:;43: Askey–Wilson | q-Racah
;32: Continuous dual q-Hahn | Continuous q-Hahn | Big q-Jacobi | q-Hahn | dual q-Hahn
;21: Al-Salam–Chihara | q-Meixner–Pollaczek | Continuous q-Jacobi | Big q-Laguerre | Little q-Jacobi | q-Meixner | Quantum q-Krawtchouk | q-Krawtchouk | Affine q-Krawtchouk | Dual q-Krawtchouk
;20/11: Continuous big q-Hermite | Continuous q-Laguerre | Little q-Laguerre | q-Laguerre | q-Bessel | q-Charlier | Al-Salam–Carlitz I | Al-Salam–Carlitz II
;10: Continuous q-Hermite | Stieltjes–Wigert | Discrete q-Hermite I | Discrete q-Hermite II
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 byabove which corresponds to the Wilson polynomials. This was ruled out in under the assumption that the are degree 1 polynomials such that
for some polynomial.