On rational transformations of linear functionals: direct problem
In this paper the authors continue their research on linearly related orthogonal polynomials (\(P_n\) and \(Q_n\)) and their functionals (\(u\) and \(v\), respectively) [\textit{M. Alfaro, F. Marcellán, A. Pena, M. L. Rezola}, J. Math. Anal. Appl. 287, No.1, 307-319 (2003; Zbl 1029.42014)]. Here they analyze the direct problem for linear functionals acting on the linear space of polynomials with complex coefficients satisfying the functional equation \(pu=qv\), where \(p(x)=x-\tilde{a}\), \(q(x)=\lambda(x-a)\). If \(u\) is a quasi-definite functional they give necessary and sufficient conditions for \(v\) to be quasi-definite. Under these conditions they characterize when the corresponding families \(P_n\) and \(Q_n\) satisfy a relation of the form \(P_n(x) + s_n P_{n-1}(x) = Q_n(x) + t_n Q_{n-1}(x)\), \(s_nt_n\neq0\).
scientific article; zbMATH DE number 2117735