ON SOBOLEV ORTHOGONALITY FOR THE GENERALIZED LAGUERRE-POLYNOMIALS

The orthogonality of the generalized Laguerre polynomials, {L(n)((alph a))(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-po sitive definite linear functional. In this work we will show that, for every value of the real parameter alpha, the generalized Laguerre pol ynomials are orthogonal with respect to a non-diagonal Sobolev inner p roduct, that is, an inner product involving derivatives. (C) 1996 Acad emic Press, Inc.