On the zeros of a class of polynomials defined by a three term recurrence relation
The polynomial \(P_{N+1} (x)\) is defined recursively by \(P_ 0(x)=0\), \(P_ 1(x) =1\), and \(\alpha_ n P_{n+1} (x)+ \alpha_{n-1} P_{n-1} (x)+ b_ n P_ n(x)=x d_ n P_ n(x)\) \((1\leq n\leq N)\); under suitable hypotheses on the coefficient sequences, it has real and simple zeros. This generalizes a result of Van Doorn and is equivalent to one of K. I. Sato. The author's proof uses the fact that the zeros of the polynomials are the eigenvalues for the problem \((AV^*+ VA+ B)f= \lambda Df\) (\(f\neq 0\)), where \(f\) is in an \(N\)-dimensional Hilbert space, \(V\) is a shift and \(A\), \(B\), \(D\) are diagonal operators.
scientific article; zbMATH DE number 584060