Connection coefficients for Laguerre--Sobolev orthogonal polynomials.
The connection problem in the framework of orthogonal polynomials is the problem of finding explicit and/or recurrence relations for the so-called connection coefficients \(c_j(n)\) of the expansion of an orthogonal family \(q_m\) in terms of another orthogonal polynomial sequence \((p_k)\) \[ q_m(x) =\sum_{j=0}^{m} c_{j}(m) p_j(x). \] In the paper the authors are interested in finding closed analytical formulas for the connection coefficients \(c_{j}(m)\) when one or both families are instances of Laguerre-Sobolev and Laguerre-Sobolev-type orthogonal polynomials. Of particular interest are the expansions of such Sobolev families in terms of the classical Laguerre polynomials as well as the corresponding inversion formulas. Finally, some open problems are posed and discussed.
scientific article; zbMATH DE number 1970828