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The associated classical orthogonal polynomials. (English) Zbl 0995.33001
Bustoz, Joaquin (ed.) et al., Special functions 2000: current perspective and future directions. Proceedings of the NATO Advanced Study Institute, Tempe, AZ, USA, May 29-June 9, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 30, 255-279 (2001).

The paper is concerned with polynomials that satisfy the three-term recurrence relation

$\begin{array}{cc}\hfill {p}_{n+1}\left(x\right)& =\left({A}_{n+c}x+{B}_{n+c}\right){p}_{n}\left(x\right)-{C}_{n+c}{p}_{n-1}\left(x\right),\phantom{\rule{4pt}{0ex}}n\in {ℕ}_{0},\hfill \\ \hfill {p}_{-1}\left(x\right)& =0,\phantom{\rule{1.em}{0ex}}{p}_{0}\left(x\right)=1,\hfill \end{array}$

where $c=0$ corresponds to a classical system, while $c\ne 0$ yields an associated system. Some examples where such polynomials occur are given in the first section. Next, the author considers the problem of finding measures of orthogonality for the polynomials; four methods (using moments, generating function, suitable special functions, and minimal soulutions, respectively) are reviewed and discussed. Finally, some particular cases are considered at some length, viz., the associated Askey-Wilson polynomials, the continuous $q$-Jacobi polynomials, the continuous $q$-ultraspherical polynomials, and the associated Wilson polynomials. There is a rather extensive bibliography.

##### MSC:
 33-02 Research monographs (special functions) 33C45 Orthogonal polynomials and functions of hypergeometric type 42C05 General theory of orthogonal functions and polynomials 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)