q-Hahn.
q-Hahn.
The q-Hahn polynomials follow from the q-Racah polynomials by the substitution
and 
in the definition (3.2.1) of the
q-Racah polynomials :
Another way to obtain the q-Hahn polynomials from the q-Racah
polynomials is by setting 
and 
in the definition
(3.2.1) :
And if we take 
, 
and 
in the
definition (3.2.1) of the q-Racah polynomials we find the
q-Hahn polynomials given by (3.6.1) in the following way :
Note that 
in each case.
Little q-Jacobi.
If we set 
in the definition (3.6.1) of the q-Hahn
polynomials and take the limit 
we find the little q-Jacobi polynomials :
where 
is defined by (3.12.1).
q-Meixner.
The q-Meixner polynomials defined by (3.13.1) can be obtained from the q-Hahn polynomials
by setting 
and 
in the definition (3.6.1) of the
q-Hahn polynomials and letting 
:
Quantum q-Krawtchouk.
The quantum q-Krawtchouk polynomials defined by (3.14.1)
simply follow from the q-Hahn polynomials by setting 
in the definition (3.6.1) of
the q-Hahn polynomials and taking the limit 
:
q-Krawtchouk.
If we set 
in the definition (3.6.1) of the q-Hahn polynomials
and then let 
we obtain the q-Krawtchouk polynomials defined by
(3.15.1) :
Affine q-Krawtchouk.
The affine q-Krawtchouk polynomials defined by (3.16.1)
can be obtained from the q-Hahn polynomials by the substitution 
and
in (3.6.1) :
