5.10 Continuous q-Jacobi --> Jacobi

If we take the limit in the definitions (3.10.1) and (3.10.14) of the continuous q-Jacobi polynomials we simply find the Jacobi polynomials defined by (1.8.1) :

and

5.10.1 Continuous q-ultraspherical / Rogers --> Gegenbauer / Ultraspherical

If we set in the definition (3.10.15) of the continuous q-ultraspherical (or Rogers) polynomials and let q tend to one we obtain the Gegenbauer (or ultraspherical) polynomials given by (1.8.15) :

5.10.2 Continuous q-Legendre --> Legendre / Spherical

The Legendre (or spherical) polynomials defined by (1.8.57) easily follow from the continuous q-Legendre polynomials given by (3.10.32) by taking the limit :

Of course, we also have