In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of
orthogonal polynomials introduced by
Richard Askey and
James A. Wilson as
q-analogs of the
Wilson polynomials.
[1] They include many of the other orthogonal polynomials in 1 variable as
special or
limiting cases, described in the
Askey scheme. Askey–Wilson polynomials are the special case of
Macdonald polynomials (or
Koornwinder polynomials) for the non-reduced
affine root system of type (C∨
1, C1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system.
They are defined by
where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.
This result can be proven since it is known that
and using the definition of the q-Pochhammer symbol
which leads to the conclusion that it equals
In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of
orthogonal polynomials introduced by
Richard Askey and
James A. Wilson as
q-analogs of the
Wilson polynomials.
[1] They include many of the other orthogonal polynomials in 1 variable as
special or
limiting cases, described in the
Askey scheme. Askey–Wilson polynomials are the special case of
Macdonald polynomials (or
Koornwinder polynomials) for the non-reduced
affine root system of type (C∨
1, C1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system.
They are defined by
where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.
This result can be proven since it is known that
and using the definition of the q-Pochhammer symbol
which leads to the conclusion that it equals