Finite perturbations of orthogonal polynomials

AbstractIn this paper we study orthogonal polynomials (pn) which arise from a given system of orthogonal polynomials (pn) by a finite perturbation of the recurrence coefficients, i.e., the recurrence coefficients (αn), (λn + 1) of (pn) respectively (αn), (λn + 1) of (pn) are related to each other by αn + l = αn + m and λn + 1 + l = λn + 1 + m for NϵN, where l and m are fixed nonnegative integers. Closed expressions for the polynomial pn in terms of pn and its associated polynomial pn − 1(1) as well as for the orthogonality measure μ of (pn) in terms of the orthogonality measure μ of (pn) are given. In fact the results are presented for the most general case when μ is an arbitrary, not necessarily positive, measure