Terminating Basic Hypergeometric Representations and Transformations for the Askey-Wilson Polynomials.

In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey-Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy.

On a generalization of the Rogers generating function.

We derive a generalization of the Rogers generating function for the continuous -ultraspherical/Rogers polynomials whose coefficient is a . From that expansion, we derive corresponding specialization and limit transition expansions for the continuous -Hermite, continuous -Legendre, Laguerre, and Chebyshev polynomials of the first kind. Using a generalized expansion of the Rogers generating function in terms of the Askey-Wilson polynomials by Ismail & Simeonov whose coefficient is a , we derive corresponding generalized expansions for the Wilson, continuous -Jacobi, and Jacobi polynomials.

Some Generating Functions for -Polynomials.

Demonstrating the striking symmetry between calculus and -calculus, we obtain -analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain -analogues for some of their generating functions. Our -generating functions are given in terms of the basic hypergeometric series , , , , , and -Pochhammer symbols.

THE POWER COLLECTION METHOD FOR CONNECTION RELATIONS: MEIXNER POLYNOMIALS.

We introduce the power collection method for easily deriving connection relations for certain hypergeometric orthogonal polynomials in the (-)Askey scheme. We summarize the full-extent to which the power collection method may be used. As an example, we use the power collection method to derive connection and connection-type relations for Meixner and Krawtchouk polynomials.