Companion linear functionals and Sobolev inner products

30 pages, no figures.-- MSC2000 codes: Primary 33C45, 42C05.MR#: MR2143522 (2006f:42022)Zbl#: Zbl 1087.42020The present paper deals with the solution of an inverse problem in the theory of orthogonal polynomials. It was motivated by a characterization result concerning sequences of polynomials orthogonal with respect to a Sobolev inner product when they can be recursively generated in terms of orthogonal polynomial sequences associated with the measure involved in the standard component. More precisely, we obtain the set of pairs of quasi–definite linear functionals such that their corresponding sequences of monic orthogonal polynomials {Pn} and {Rn} are related by a differential expression $$ \frac{R'_{n+1}(x)}{n+1}+b_n\frac{R'_n(x)}{n}=P_n(x)+a_nP_{n-1}(x) \tag 2$$ where $ b_n\ne0$ for every n ∈ N.The work of the authors has been supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología of Spain, under grant BFM 2003–06335–C03–02 and INTAS Research Network NeCCA INTAS 03–31–6637.Publicad