On second kind polynomials associated with rational transformations of linear functionals
This paper is concerned with the study of rational transformations of linear functionals. This is a very well written paper which describes the following situation. For a regular and hermitian functional \(u\), it is considered a new linear functional \(\mathcal{L}\) defined by means of the following relations with \(u\): \[ \lambda (z-\beta)\mathcal{L}=(z-\alpha)u, \] with \(\alpha, \beta, \lambda \in \mathbb{C}\) and \(\lambda \neq 0.\) A necessary and sufficient condition for the regularity of \(\mathcal{L}\) is obtained and the positive definite character of \( \mathcal{L}\) is also studied. Some linear relations linking the sequences of orthogonal polynomials related to \(\mathcal{L}\) and \(u\) are obtained as well as an explicit representation for the orthogonal polynomial sequences of second kind related to \(\mathcal{L}\). Finally, the particular case \(\alpha=\beta\) is analyzed. In this last situation \(\mathcal{L}\) becomes in the modification of \(u\) by addition of a Dirac delta.
scientific article; zbMATH DE number 5577101