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Orthogonal Laurent polynomials. (English) Zbl 0604.33004

The authors consider sequences of Laurent polynomials {Q k } 0 where

Q 2n =α -n (2n) x -n +···+α n (2n) x n ,α n (2n) 0,
Q 2n+1 =α -n-1 (2n+1) x -n-1 +···+α n (2n+1) x n ,α -n-1 (2n+1) 0·

A linear functional L determined by {Q k } 0 is defined as L(Q k Q n )=0 for kn and 0 for k=n. Propositions and theorems are proved that link the existence of L, {Q k } and three- term linear recurrence relations for the Q’s and thereby general T- fractions.

A sequence of lacunary Laurent polynomials {P n } 0 is introduced orthogonal with relation to L 1 where L 1 (x 2n+1 )=0, L 1 (x 2n )=L(x n ). Applications are made to ratios of hypergeometric functions. Orthogonal Laurent polynomials were introduced in W. B. Jones and W. J. Thron [Analytic theory of continued fractions, Proc. Sem. Workshop, Loen/Norw. 1981, Lect. Notes Math. 932, 4-37 (1982; Zbl 0508.30008)] and further developed in O. Njåstad and W. J. Thron, Skr., K. Nor. Vidensk. Selsk. (1983).

Reviewer: A.Magnus

MSC:
33C45Orthogonal polynomials and functions of hypergeometric type
33C05Classical hypergeometric functions, 2 F 1