Continuous dual q-Hahn.
Continuous dual q-Hahn.
The continuous dual q-Hahn polynomials defined by (3.3.1) simply follow
from the Askey-Wilson polynomials given by (3.1.1) by setting
d = 0 in (3.1.1) :
Continuous q-Hahn.
The continuous q-Hahn polynomials defined by (3.4.1) can be obtained
from the Askey-Wilson polynomials given by (3.1.1) by the substitutions
, 
, 
,
and 
:
Big q-Jacobi.
The big q-Jacobi polynomials defined by (3.5.1) can be obtained from the
Askey-Wilson polynomials by setting 
, 
,
and 
in
defined by (3.1.1) and then taking the limit 
:
Continuous q-Jacobi.
If we take 
, 
,
and 
in
the definition (3.1.1) of the Askey-Wilson polynomials and
change the normalization we find the continuous q-Jacobi polynomials given
by (3.10.1) :
In [345] M. Rahman takes 
,
, 
and
to obtain after a change of normalization the
continuous q-Jacobi polynomials defined by (3.10.14) :
As was pointed out in section 0.6 these two q-analogues of the Jacobi polynomials are not really different, since they are connected by the quadratic transformation
Continuous q-ultraspherical / Rogers.
If we set 
, 
,
and 
in the definition
(3.1.1) of the Askey-Wilson polynomials and change the
normalization we obtain the continuous q-ultraspherical (or Rogers)
polynomials defined by (3.10.15). In fact we have :
