Blumenthal's theorem for Laurent orthogonal polynomials
The authors study polynomials which satisfy a three-term recurrence relation \(B_n(x)=(x-\beta_n)B_{n-1}(x)-\alpha_n xB_{n-2}(x)\) with positive coefficients \(\alpha_{n+1}, \beta_n\) and show that their zeros are eigenvalues of structured Hessenberg matrices. The case when \(\alpha_n \to\alpha\) and \(\beta_n \to\beta\) is analysed in more detail. Now the support of the Laurent orthogonality measure is an interval and the Laurent version of Blumenthal's theorem holds. The authors also prove the ratio asymptotics and study the asymptotic behavior for the logarithmic derivative for the polynomials \(B_n\) in case \(\alpha>0\) and \(\beta>0\).
scientific article; zbMATH DE number 1891494
scientific article