4.2 q-Racah

q-Racah Big q-Jacobi.

  
The big q-Jacobi polynomials defined by (3.5.1) can be obtained from the q-Racah polynomials by setting in the definition (3.2.1) :

q-Racah 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.

q-Racah Dual q-Hahn.

  
To obtain the dual q-Hahn polynomials from the q-Racah polynomials we have to take and in (3.2.1) :

with

We may also take and in (3.2.1) to obtain the dual q-Hahn polynomials from the q-Racah polynomials :

with

And if we take , and in the definition (3.2.1) of the q-Racah polynomials we find the dual q-Hahn polynomials given by (3.7.1) in the following way :

with

q-Racah q-Krawtchouk.

  
The q-Krawtchouk polynomials defined by (3.15.1) can be obtained from the q-Racah polynomials by setting , and in the definition (3.2.1) of the q-Racah polynomials :

Note that in this case.

q-Racah Dual q-Krawtchouk.

  
The dual q-Krawtchouk polynomials defined by (3.17.1) easily follow from the q-Racah polynomials given by (3.2.1) by using the substitutions , and :

Note that