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  to cover basic orthogonal polynomials.

Askey scheme for hypergeometric orthogonal polynomials
give the following version of the Askey scheme:

Here $${}_pF_q(n)$$ indicates a hypergeometric series representation with $$n$$ parameters
 * $${}_4F_3(4)$$: Wilson | Racah
 * $${}_3F_2(3)$$: Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn
 * $${}_2F_1(2)$$: Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk
 * $${}_2F_0(1)\ \ / \ \ {}_1F_1(1)$$: Laguerre | Bessel | Charlier
 * $${}_2F_0(0)$$: Hermite

Askey scheme for basic hypergeometric orthogonal polynomials
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
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$$ which corresponds to the Wilson polynomials. This was ruled out in under the assumption that the $$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)$$.