4.17 Dual q-Krawtchouk

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

Dual q-Hahn Dual q-Krawtchouk.

  
The dual q-Krawtchouk polynomials defined by (3.17.1) can be obtained from the dual q-Hahn polynomials by setting in (3.7.1) and then letting :

Dual q-Krawtchouk Al-Salam-Carlitz I.

  
If we set in the definition (3.17.1) of the dual q-Krawtchouk polynomials and take the limit we simply obtain the Al-Salam-Carlitz I polynomials given by (3.24.1) :

Note that .