4.10 Continuous q-Jacobi

Askey-Wilson 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-Jacobi Continuous q-Laguerre.

   
The continuous q-Laguerre polynomials given by (3.19.1) and (3.19.15) follow simply from the continuous q-Jacobi polynomials defined by (3.10.1) and (3.10.14) respectively by taking the limit :

and

4.10.1 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 :

q-Meixner-Pollaczek Continuous q-ultraspherical / Rogers.

    
If we take and in the definition (3.9.1) of the q-Meixner-Pollaczek polynomials we obtain the continuous q-ultraspherical (or Rogers) polynomials given by (3.10.15) :

Continuous q-ultraspherical / Rogers Continuous q-Hermite.

     
The continuous q-Hermite polynomials defined by (3.26.1) can be obtained from the continuous q-ultraspherical (or Rogers) polynomials given by (3.10.15) by taking the limit . In fact we have