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 edit

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

${}_{4}F_{3}(4)$: Wilson | Racah
${}_{3}F_{2}(3)$: Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn
${}_{2}F_{1}(2)$: Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk
${}_{2}F_{0}(1)\ \ /\ \ {}_{1}F_{1}(1)$: Laguerre | Bessel | Charlier
${}_{2}F_{0}(0)$: Hermite

Here ${}_{p}F_{q}(n)$ indicates a hypergeometric series representation with $n$ parameters

Askey scheme for basic hypergeometric orthogonal polynomials edit

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

Completeness edit

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

p_{n}(x)={}_{q+1}F_{q}\left({\begin{array}{c}-n,n+\mu ,a_{1}(x),\dots ,a_{q-1}(x)\\b_{1},\dots ,b_{q}\end{array}};1\right)

above $q=3$ which corresponds to the Wilson polynomials. This was ruled out in Cheikh, Lamiri & Ouni (2009) under the assumption that the $a_{i}(x)$ are degree 1 polynomials such that

\prod _{i=1}^{q-1}(a_{i}(x)+r)=\prod _{i=1}^{q-1}a_{i}(x)+\pi (r)

for some polynomial $\pi (r)$ .

References edit

Andrews, George E.; Askey, Richard (1985), "Classical orthogonal polynomials", in Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A. (eds.), Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984., Lecture Notes in Math., vol. 1171, Berlin, New York: Springer-Verlag, pp. 36–62, doi:10.1007/BFb0076530, ISBN 978-3-540-16059-5, MR 0838970
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 0783216
Cheikh, Y. Ben; Lamiri, I.; Ouni, A. (2009), "On Askey-scheme and d-orthogonality, I: A characterization theorem", Journal of Computational and Applied Mathematics, 233: 621–629
Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, vol. 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., vol. 1329, Berlin, New York: Springer-Verlag, pp. 46–72, doi:10.1007/BFb0083353, ISBN 978-3-540-19489-7, MR 0973421
Labelle, Jacques (1985), "Tableau d'Askey", in Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A. (eds.), Polynômes Orthogonaux et Applications. Proceedings of the Laguerre Symposium held at Bar-le-Duc, Lecture Notes in Math., vol. 1171, Berlin, New York: Springer-Verlag, pp. xxxvi–xxxvii, doi:10.1007/BFb0076527, ISBN 978-3-540-16059-5, MR 0838967