Zeros of polynomials embedded in an orthogonal sequence
\textit{B. Wendroff} [Proc.\ Am.\ Math.\ Soc.\ 12, 554--555 (1961; Zbl 0099.05601)] proved the following theorem. Let be given real points \[ x_{1,n+1}<x_{1,n}<x_{2,n+1}<x_{2,n}<\cdots<x_{n,n+1}<x_{n,n}<x_{n+1,n+1}. \] Then the polynomials \(p_{n}(x)=\prod_{k=1}^n(x-x_{k,n})\) and \(p_{n+1}(x)=\prod_{k=1}^{n+1}(x-x_{k,n+1})\) can be embedded in an infinite sequence \((p_n)_n\) of monic polynomials which are orthogonal with respect to some positive Borel measure on \(\mathbb R\). Any monic orthogonal sequence \((p_n)_n\) must satisfy a recurrence relation of the form \[ p_{n+1}(x)=(x-b_n)p_n(x)-\lambda_n p_{n-1},\quad n \in \mathbb N \tag{1} \] where \(p_0(x)=1\), \(p_{-1}(x)=0\), \(\lambda_n>0\), and \(b_n \in \mathbb R\). It follows that \(p_1,p_2,\ldots,p_{n-1}\) are uniquely determined by \(p_n\) and \(p_{n+1}\). In contrast, the polynomials \(p_{k}\), \(k\geq n+2\), are constructed successively by using (1) and choosing constants \(b_k\) and \(\lambda_k\). The authors investigate how the location of the zeros of \(p_{n+2}\) depends on the choices of \(b_{n+1}\) and \(\lambda_{n+1}\). This applies recursively to \(p_{n+3}\), \(p_{n+4}\), etc.
scientific article; zbMATH DE number 5935048
scientific article