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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

p n+1 (x)=(A n+c x+B n+c )p n (x)-C n+c p n-1 (x),n 0 ,p -1 (x)=0,p 0 (x)=1,

where c=0 corresponds to a classical system, while c0 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:
33-02Research 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.)