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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 $$\align p_{n+1}(x) & =(A_{n+c}x+ B_{n+c})p_n(x)-C_{n+c} p_{n-1}(x),\ n\in\bbfN_0,\\ p_{-1}(x) & =0,\quad p_0(x)=1, \endalign$$ 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. For the entire collection see [Zbl 0969.00053].
33-02Research monographs (special functions)
33C45Orthogonal polynomials and functions of hypergeometric type
42C05General theory of orthogonal functions and polynomials
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)