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Zbl 0604.33004
Hendriksen, E.; van Rossum, H.
Orthogonal Laurent polynomials. (English)
[J] Indag. Math. 48, 17-36 (1986). ISSN 0019-3577

The authors consider sequences of Laurent polynomials $\{Q\sb k\}\sb 0\sp{\infty}$ where $$ Q\sb{2n}=\alpha\sb{-n}\sp{(2n)} x\sp{- n}+...+\alpha\sb n\sp{(2n)} x\sp n,\quad \alpha\sb n\sp{(2n)}\ne 0, $$ $$ Q\sb{2n+1}=\alpha\sb{-n-1}\sp{(2n+1)} x\sp{-n-1}+...+\alpha\sb n\sp{(2n+1)} x\sp n,\quad \alpha\sb{-n-1}\sp{(2n+1)}\ne 0. $$ A linear functional L determined by $\{Q\sb k\}\sb 0\sp{\infty}$ is defined as $L(Q\sb kQ\sb n)=0$ for $k\ne n$ and $\ne 0$ for $k=n$. Propositions and theorems are proved that link the existence of L, $\{Q\sb k\}$ and three- term linear recurrence relations for the Q's and thereby general T- fractions. \par A sequence of lacunary Laurent polynomials $\{P\sb n\}\sb 0\sp{\infty}$ is introduced orthogonal with relation to $L\sb 1$ where $L\sb 1(x\sp{2n+1})=0$, $L\sb 1(x\sp{2n})=L(x\sp n)$. Applications are made to ratios of hypergeometric functions. Orthogonal Laurent polynomials were introduced in {\it W. B. Jones} and {\it 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 {\it O. Njåstad} and {\it W. J. Thron}, Skr., K. Nor. Vidensk. Selsk. (1983).
[A.Magnus]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
33C05 Classical hypergeometric functions

Keywords: Laurent polynomials

Citations: Zbl 0508.30008

Cited in: Zbl 0893.33004 Zbl 0659.33004

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