Diagonal orthogonal polynomial sequences
Let \((p_n)_{n\geq 0}\) be a sequence of polynomials, \(\phi(x)\) a polynomial of degree \(t\) and \(s\geq 0\) an integer such that \[ \phi(x)p_n(x)=\sum_{k=n-s}^{n+t} \lambda_{n,k} p_{k+1}'(x),\quad n\geq 0, \] where \(\lambda_{n,k}\) are some constants such that \(\lambda_{n,n-s}\neq 0\) for \(n\geq s\). Let pose the following problem: To characterize all orthogonal polynomial sequences satisfying the above relation. The case \(\phi(x)=1\), i.e., \(t=0\), leads to the classical orthogonal polynomials (see \textit{F. Marcellán, A. Branquinho}, and \textit{J. Petronilho}, ``Classical orthogonal polynomials: A functional approach'', Acta Appl. Math. 34, 283-303 (1994; Zbl 0793.33009)]). The aim of this paper is the detailed study of the case \(t=1\) (\(\phi(x)=x+1\)). In fact, the authors find that the arising sequences are the classical forms and some perturbations of them (e.g., addition of delta Dirac measures, multiplication by first degree polynomials).
scientific article; zbMATH DE number 1757331