We study special values for the continuous $q$-Jacobi polynomials and present applications of these special values which arise from bilinear generating functions, and in particular the Poisson kernel for these polynomials.
How do the special values of the continuous $q$-Jacobi polynomials, derived from bilinear generating functions and the Poisson kernel, contribute to the understanding of the relationship between the Askey-Wilson polynomials and the continuous $q$-Jacobi polynomials? Can you provide a specific example to illustrate this connection?
In fact, the special values of the continuous q-Jacobi polynomials are not derived from bilinear generating functions. They come directly from the basic hypergeometric representations of these polynomials which are special cases of teh Askey-Wilson polynomials. The connection with bilinear generating functions is that when you use of these special values in the bilinear generating functions, you automatically produce generating functions and arbitrary argument transformations for nonterminating basic hypergeometric functions, which is quite interesting and have broad applications. We apply these special values specifically to the Poisson kernel which is a very special bilinear generating functions for these polynomials. Here I present examples of the special values, an example of generating functions and an example of a nonterminating transformation.
For some reason the voice says "equi-Jacobi polynomials" when it should be "q-Jacobi polynomials"