On Sobolev orthogonality for the generalized Laguerre polynomials

The orthogonality of the generalized Laguerre polynomials, {L(n)((alpha))(x)} (n greater than or equal to 0), is a well known fact when the parameter alpha is a real number but not a negative integer. In fact, for -1 <alpha, they are orthogonal on the interval [0 + infinity) with respect to the weight function rho(x) = x(alpha)e(-x), and for alpha < -1, but not an integer, they are orthogonal with respect to a non-positive definite linear functional. In this work we will show that, for every value of the real parameter alpha, the generalized Laguerre polynomials are orthogonal with respect to a non-diagonal Sobolev inner product, that is, an inner product involving derivatives. (C) 1996 Academic Press, Inc.