Recurrence coefficients of orthogonal polynomials with respect to some self-similar singular distributions
Monic orthogonal polynomials \(P_ n\) with orthogonality measure on \({\mathbf R}\) satisfy a 3-term recurrence relation \[ P_ n(x)= (x- c_ n) P_{n- 1}(x)- \lambda_ n P_{n- 2}(x)\quad (n\geq 1). \] The limit behavior of the arithmetic and geometric means of the coefficients \(c_ n\) and \(\lambda_ n\) is investigated. It is not assumed that the orthogonality measure is absolutely continuous, but regular \(n\)th root asymptotic behavior of the orthogonal polynomials \(P_ n\) must be guaranteed. Some examples of self-similar distributions satisfying this condition are given.
scientific article; zbMATH DE number 739098