Definition.
where
and
Since
it is clear that is a polynomial of degree n in .
Orthogonality.
where
and
This implies
Recurrence relation.
where
Normalized recurrence relation.
where
q-Difference equation.
where
and as below. This q-difference equation can also be written in the form
where
and
Forward shift operator.
or equivalently
Backward shift operator.
or equivalently
where
Rodrigues-type formula.
where
Generating functions. For we have
Remarks. The Askey-Wilson polynomials defined by (3.1.1) and the q-Racah polynomials given by (3.2.1) are related in the following way. If we substitute , , , and in the definition (3.2.1) of the q-Racah polynomials we find :
and
If we change q by we find
where
References. [13], [26], [31], [62], [64], [67], [117], [118], [160], [188], [190], [193], [218], [245], [279], [323], [331], [346].