Connection coefficients for Laguerre–Sobolev orthogonal polynomials

Laguerre–Sobolev polynomials are orthogonal with respect to an inner product of the form 〈p,q〉S = ∫∞ 0 p(x)q(x)x αe−x dx + λ∫∞ 0 p′(x)q′(x)dμ(x), where α > −1, λ 0, and p,q ∈ P, the linear space of polynomials with real coefficients. If dμ(x) = xαe−x dx, the corresponding sequence of monic orthogonal polynomials {Q n (x)} has been studied by Marcellán et al. (J. Comput. Appl. Math. 71 (1996) 245–265), while if dμ(x) = δ(x)dx the sequence of monic orthogonal polynomials {L n (x;λ)} was introduced by Koekoek and Meijer (SIAM J. Math. Anal. 24 (1993) 768–782). For each of these two families of Laguerre–Sobolev polynomials, here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre–Sobolev polynomials, and the connection problem relating two families of Laguerre– Sobolev polynomials with different parameters, are also considered.  2003 Elsevier Inc. All rights reserved.