A note on the interlacing of zeros and orthogonality
The author considers a sequence \(\{t_n\}_{n=0}^{\infty}\) of real monic polynomials, with \(\deg(t_n)=n\) for each \(n\), satisfying the property that the zeros of \(t_n\) are real and simple and two consecutive polynomials have no common zeros for any \(n \in \mathbb{N}.\) It is also assumed that the ratio \(\frac{t_{n+1}}{t_{n-1}}\) takes the same value at the zeros of \(t_n\). Under these assumptions, two necessary and sufficient conditions for the orthogonality of \(\{t_n\}\) are obtained. The conditions are given in terms of the interlacing properties of the zeros of \(t_n\) and \(t_{n+1}\) and in terms of the positivity of the ratios above introduced.
orthogonality of a monic polynomial sequence
scientific article; zbMATH DE number 5655034