** Definition.**

The Askey-Wilson polynomials are *q*-analogues of the Wilson polynomials given by (1.1.1).

** Orthogonality.** If *a,b,c,d* are real, or occur in complex conjugate
pairs if complex, and

, then we have the following orthogonality relation

where

with

and

If *a* > 1 and *b,c,d* are real or one is real and the other two are complex conjugates,

and the pairwise products of *a,b,c* and *d* have
absolute value less than one, then we have another orthogonality relation
given by :

where and are as before,

and

** Recurrence relation.**

where

and

** Normalized recurrence relation.**

where

** q-Difference equation.**

where

and

If we define

then the *q*-difference equation can also be written in the form

where

** Forward shift operator.**

or equivalently

** Backward shift operator.**

or equivalently

** Rodrigues-type formula.**

** Generating functions.**

** Remarks.**
The *q*-Racah polynomials defined by (3.2.1) and the Askey-Wilson
polynomials given by (3.1.1) are related in the following way.
If we substitute , ,
, and
in the definition (3.1.1)
of the Askey-Wilson polynomials we find :

where

If we change *q* by we find

** References.**
[13], [31], [43],
[58], [64], [67],
[69], [70], [96],
[97], [191], [193],
[203], [204], [218],
[224], [226], [230],
[231], [234], [238],
[242], [249], [256],
[259], [281], [282], [293],
[318], [322], [323],
[324], [328], [346],
[347], [349], [350], [352],
[353], [355], [359],
[371], [389], [400].

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