The strong Chebyshev distribution and orthogonal Laurent polynomials
This is a worked out example of a sequence of orthogonal Laurent polynomials, based on the Chebyshev polynomials. The strong Chebyshev distribution is a (symmetric) weight on \([-b,-a] \cup [a,b]\) for which all the moments exist, even for negative indices. It is obtained from the Chebyshev weight \(1/\sqrt{1-x^2}\) on \([-1,1]\) by a quadratic transformation. The corresponding Chebyshev Laurent polynomials are obtained from the Chebyshev polynomials of the first kind by using a rational change of variable. This allows the authors to explicitly find the orthogonality, the recurrence relation, and closed form expressions. The (uniform) convergence of the continued fraction associated with these Laurent polynomials is investigated. Finally, the zeros of the Chebyshev-Laurent polynomials are given explicitly. \textit{B. A. Hagler, W. B. Jones} and \textit{W. J. Thron} [Lect. Notes Pure Appl. Math. 199, 187-208 (1998)] have used the same idea to find orthogonal Laurent polynomials of Jacobi, Hermite and Laguerre types.
scientific article; zbMATH DE number 1197037
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