UPWARD EXTENSION OF THE JACOBI MATRIX FOR ORTHOGONAL POLYNOMIALS

Orthogonal polynomials on the real line always satisfy a three-term re currence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix r new rows and columns, so that the origin al Jacobi matrix is shifted downward. The r new rows and columns conta in 2r new parameters and the newly obtained orthogonal polynomials thu s correspond to an upward extension of the Jacobi matrix. We give an e xplicit expression of the new orthogonal polynomials in terms of the o riginal orthogonal polynomials, their associated polynomials, and the 2r new parameters, and we give a fourth order differential equation fo r these new polynomials when the original orthogonal polynomials are c lassical. Furthermore we show how the orthogonalizing measure for thes e new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associa ted polynomials are again Jacobi polynomials. (C) 1996 Academic Press, Inc.