Mizan Rahman, Carleton University, Canada
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Lecture 1
The associated classical orthogonal polynomials

Abstract

The associated orthogonal polynomials $\{p_n(x;c)\}$ are defined by the 3-term recurrence relation with coefficients $A_n,\, B_n,\, C_n$ for $\{p_n(x)\}$ with $c=0$, replaced by $A_{n+c},\, B_{n+c}$ and $C_{n+c}$, $c$ being the associated parameter.  Starting with examples where such polynomials occur in a natural way some of the well-known theories of how to determine their measures of orthogonality are discussed.  The highest level of the family of classical orthogonal polynomials, namely, the associated Askey-Wilson polynomials which were studied at length by Ismail and Rahman in 1991 is reviewed with special reference to various connected results that exist in the literature.

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Lecture 2
On an inverse of the Askey-Wilson operator: the indefinite integral

Abstract

Following the work of Ismail, Rahman and Zhang on the diagonalization of the inverse Askey-Wilson operator, an exact evaluation is made of the kernel of the corresponding integral operator in terms of ratios of theta functions. This evaluation enables us to link up the inverse of the Asley-Wilson operator with the standard indefinite integral and the analogue of an arbitrary constant in this context. The positivity of the integral operator is established by use  of some theta function identities.

View Lecture Notes on q-Beta Integrals and Summation Theorems  (graduate seminar taught at Arizona State University during the Spring of 1999) ps, pdf.