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Orthogonal Laurent polynomials. (English) Zbl 0604.33004

The authors consider sequences of Laurent polynomials ${\left\{{Q}_{k}\right\}}_{0}^{\infty }$ where

${Q}_{2n}={\alpha }_{-n}^{\left(2n\right)}{x}^{-n}+···+{\alpha }_{n}^{\left(2n\right)}{x}^{n},\phantom{\rule{1.em}{0ex}}{\alpha }_{n}^{\left(2n\right)}\ne 0,$
${Q}_{2n+1}={\alpha }_{-n-1}^{\left(2n+1\right)}{x}^{-n-1}+···+{\alpha }_{n}^{\left(2n+1\right)}{x}^{n},\phantom{\rule{1.em}{0ex}}{\alpha }_{-n-1}^{\left(2n+1\right)}\ne 0·$

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

A sequence of lacunary Laurent polynomials ${\left\{{P}_{n}\right\}}_{0}^{\infty }$ is introduced orthogonal with relation to ${L}_{1}$ where ${L}_{1}\left({x}^{2n+1}\right)=0$, ${L}_{1}\left({x}^{2n}\right)=L\left({x}^{n}\right)$. 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:
 33C45 Orthogonal polynomials and functions of hypergeometric type 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$
##### Keywords:
Laurent polynomials