Some functions that generalize the Krall-Laguerre polynomials

Let L(alpha) be the (semi-infinite) tridiagonal matrix associated with the three-term recursion relation satisfied by the Laguerre polynomials, with weight function 1/Gamma(alpha+1)z(alpha)e(-z), alpha>-1, on the interval [0, infinity]. We show that, when alpha is a positive integer, by performing at most alpha successive Darboux transformations from L(alpha), we obtain orthogonal polynomials on [0, infinity] with 'weight distribution' 1/Gamma(alpha-k+1)z(alpha-k)e(-z) + Sigma(j=1)(k) s(j)delta((k-j))(z), with 1 less than or equal to k less than or equal to alpha. We prove that, as a consequence of the rational character of the Darboux factorization, these polynomials are eigenfunctions of a (finite order) differential operator. Our construction calls for a natural bi-infinite extension of these results with polynomials replaced by functions, of which the semi-infinite case is a limiting situation. (C) 1999 Elsevier Science B.V. All rights reserved.