In mathematics, the Askey scheme is a way of organizing orthogonal polynomials

Orthogonal polynomials

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...

 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:

₄F₃: Wilson

Wilson polynomials

In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials....

 Racah

Racah polynomials

In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients....

₃F₂: Continuous dual Hahn

Continuous dual Hahn polynomials

In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials...

 Continuous Hahn

Continuous Hahn polynomials

In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials...

 Hahn

Hahn polynomials

In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Chebyshev in 1875 and rediscovered by . The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner...

 dual Hahn
₂F₁: Meixner–Pollaczek Jacobi

Jacobi polynomials

In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...

 Pseudo Jacobi

Pseudo Jacobi polynomials

In mathematics, the Pseudo Jacobi polynomials are a finite sequence Pn of orthogonal polynomials that can be expressed in terms of Jacobi polynomials of imaginary argument. They were introduced by ....

 Meixner Krawtchouk
₂F₀/₁F₁: Laguerre

Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....

 Bessel

Bessel polynomials

In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions...

 Charlier

Charlier polynomials

In mathematics, Charlier polynomials are a family of orthogonal polynomials introduced by Carl Charlier....

₁F₀: Hermite

Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...

Askey scheme for basic hypergeometric orthogonal polynomials

give the following scheme for basic hypergeometric orthogonal polynomials:

₄F₃: Askey–Wilson

Askey–Wilson polynomials

In mathematics, the Askey–Wilson polynomials are a family of orthogonal polynomials introduced by as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme...

 q-Racah

Q-Racah polynomials

In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...

₃F₂: Continuous dual q-Hahn

Continuous dual q-Hahn polynomials

In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Continuous q-Hahn

Continuous q-Hahn polynomials

In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Big q-Jacobi

Big q-Jacobi polynomials

In mathematics, the big q-Jacobi polynomials Pn, introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme...

 q-Hahn

Q-Hahn polynomials

In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 dual q-Hahn

Dual q-Hahn polynomials

In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

₂F₁: Al-Salam–Chihara

Al-Salam–Chihara polynomials

In mathematics, the Al-Salam–Chihara polynomials Qn are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...

 q-Meixner–Pollaczek

Q-Meixner–Pollaczek polynomials

In mathematics, the q-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Continuous q-Jacobi

Continuous q-Jacobi polynomials

In mathematics, the continuous q-Jacobi polynomials P, introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Big q-Laguerre

Big q-Laguerre polynomials

In mathematics, the big q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Little q-Jacobi

Little q-Jacobi polynomials

In mathematics, the little q-Jacobi polynomials pn are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...

 q-Meixner

Q-Meixner polynomials

In mathematics, the q-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Quantum q-Krawtchouk

Quantum q-Krawtchouk polynomials

In mathematics, the quantum q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 q-Krawtchouk

Q-Krawtchouk polynomials

In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties....

 Affine q-Krawtchouk

Affine q-Krawtchouk polynomials

In mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. give a detailed list of their properties.-Definition:...

 Dual q-Krawtchouk

Dual q-Krawtchouk polynomials

In mathematics, the dual q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

₂F₀/₁F₁: Continuous big q-Hermite

Continuous big q-Hermite polynomials

In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Continuous q-Laguerre

Continuous q-Laguerre polynomials

In mathematics, the continuous q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Little q-Laguerre

Little q-Laguerre polynomials

In mathematics, the little q-Laguerre polynomials pn or Wall polynomials Wn are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to a continued fraction studied by...

 q-Laguerre

Q-Laguerre polynomials

In mathematics, the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by...

 q-Bessel

Q-Bessel polynomials

In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 q-Charlier

Q-Charlier polynomials

In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Al-Salam–Carlitz I

Al-Salam–Carlitz polynomials

In mathematics, Al-Salam–Carlitz polynomials U and V are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...

 Al-Salam–Carlitz II

Al-Salam–Carlitz polynomials

In mathematics, Al-Salam–Carlitz polynomials U and V are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...

₁F₀: Continuous q-Hermite

Continuous q-Hermite polynomials

In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...

 Stieltjes–Wigert Discrete q-Hermite I

Discrete q-Hermite polynomials

In mathematics, the discrete q-Hermite polynomials are two closely related families hn and ĥn of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...

 Discrete q-Hermite II

Discrete q-Hermite polynomials

In mathematics, the discrete q-Hermite polynomials are two closely related families hn and ĥn of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...