Askey scheme

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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[edit]

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[edit]

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

4\phi3
Askey–Wilson q-Racah
3\phi2
Continuous dual q-Hahn Continuous q-Hahn Big q-Jacobi q-Hahn dual q-Hahn
2\phi1
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\phi0/1\phi1
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\phi0
Continuous q-Hermite Stieltjes–Wigert Discrete q-Hermite I Discrete q-Hermite II

References[edit]