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 Andrews & Askey (1985), the Askey scheme was first drawn by Labelle (1985) and by Askey and Wilson (1985), and has since been extended by Koekoek & Swarttouw (1998) and Koekoek, Lesky & Swarttouw (2010) to cover basic orthogonal polynomials.

Askey scheme for hypergeometric orthogonal polynomials

Koekoek, Lesky & Swarttouw (2010, p.183) give the following version of the Askey scheme:

₄F₃

Wilson | Racah

₃F₂

Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn

₂F₁

Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk

₂F₀/₁F₁

Laguerre | Bessel | Charlier

₁F₀

Hermite

Askey scheme for basic hypergeometric orthogonal polynomials

Koekoek, Lesky & Swarttouw (2010, p.413) give the following scheme for basic hypergeometric orthogonal polynomials:

₄$\phi$₃

Askey–Wilson | q-Racah

₃$\phi$₂

Continuous dual q-Hahn | Continuous q-Hahn | Big q-Jacobi | q-Hahn | dual q-Hahn

₂$\phi$₁

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

₂$\phi$₀/₁$\phi$₁

Continuous big q-Hermite | Continuous q-Laguerre | Little q-Laguerre | q-Laguerre | q-Bessel | q-Charlier | Al-Salam–Carlitz I | Al-Salam–Carlitz II

₁$\phi$₀

Continuous q-Hermite | Stieltjes–Wigert | Discrete q-Hermite I | Discrete q-Hermite II