Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices Original Research Article

Chain sequences are positive sequences {an} of the forman=gn(1−gn−1) for a nonnegative sequence {gn}. This concept was introduced by Wall in connection with continued fractions. In his monograph on orthogonal polynomials, Chihara conjectured that ifan⩾14 for eachnthen ∑ (an−14)⩽14. We prove this conjecture and give other precise estimates foran. We also characterize the chain sequences {an} whose terms are greater than 14. We show connections to Jacobi matrices and orthogonal polynomials. In particular, we characterize the maximal chain sequences in terms of integrability properties of the spectral measure of the associated Jacobi matrix.