# .

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:

$$_4F_3$$
Wilson | Racah
$$_3F_2$$
Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn
$$_2F_1$$
Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk
$$_2F_0/_1F_1$$
Laguerre | Bessel | Charlier
$$_1F_0$$
Hermite

Askey scheme for basic hypergeometric orthogonal polynomials

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

$$_4\phi_3$$

$$_3\phi_2$$

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

References

Andrews, George E.; Askey, Richard (1985), "Classical orthogonal polynomials", in Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A., Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984., Lecture Notes in Math. 1171, Berlin, New York: Springer-Verlag, pp. 36–62, doi:10.1007/BFb0076530, ISBN 978-3-540-16059-5, MR 838970
Askey, Richard; Wilson, James (1985), "Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials", Memoirs of the American Mathematical Society 54 (319): iv+55, doi:10.1090/memo/0319, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR 783216
Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H. (1988), "Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials", Orthogonal polynomials and their applications (Segovia, 1986), Lecture Notes in Math. 1329, Berlin, New York: Springer-Verlag, pp. 46–72, doi:10.1007/BFb0083353, ISBN 978-3-540-19489-7, MR 973421
Labelle, Jacques (1985), "Tableau d'Askey", in Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A., Polynômes Orthogonaux et Applications. Proceedings of the Laguerre Symposium held at Bar-le-Duc, Lecture Notes in Math. 1171, Berlin, New York: Springer-Verlag, pp. xxxvi–xxxvii, doi:10.1007/BFb0076527, ISBN 978-3-540-16059-5, MR 838967