The one-quarter class of orthogonal polynomials (Q1178578)

Some important results about the spectrum of orthogonal polynomials, defined by a recurrence relation \[ P_ n(x)=(x-c_ n)P_{n-1}(x)- \lambda_ n P_{n-2}(x), \] are reviewed. Special attention is given to the parameters \(\sigma\) and \(\tau\) which for determined moment problems correspond to the smallest and largest limit points of the spectrum. After a short survey about what is known for a bounded spectral interval, the author takes a closer look at the case of an unbounded spectral interval. The ratio \(L(n)=\lambda_{n+1}/(c_ n c_{n+1})\) is crucial and the results very strongly depend on whether \(\limsup L(n)<1/4\) or \(\liminf L(n)>1/4\), hence the title of the paper. No proofs are given but the references are detailed, historically relevant and should be sufficient for the interested reader for further study.