Orthogonal polynomials for the Sobolev inner product
are investigated. An interesting notion of coherent pairs of Borel measures and is introduced. If are the orthogonal polynomials with respect to the measure , then is a coherent pair when there exist non-zero constants such that
It is shown that for a coherent pair (,) the Sobolev orthogonal polynomials can be expanded in terms of the orthogonal polynomials in such a way that the expansion coefficients (except the last) are independent of n and themselves orthogonal polynomials in the variable . Several examples are worked out and an application involving the evaluation of Sobolev-Fourier coefficients is given.