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. 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", "C"1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system.
They are defined by 
formula_1
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.
Proof.
This result can be proven since it is known that
formula_2
and using the definition of the "q"-Pochhammer symbol
formula_3
which leads to the conclusion that it equals
formula_4
References.
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