Some functions that generalize the Krall-Laguerre polynomials

Let L(α) be the (semi-infinite) tridiagonal matrix associated with the three-term recursion relation satisfied by the Laguerre polynomials, with weight function 1Г(α+1)Zxe-z, α > − 1, on the interval [0,∞[. We show that, when α is a positive integer, by performing at most α successive Darboux transformations from L(α), we obtain orthogonal polynomials on [0,∞[ with ‘weight distribution’ 1Г(α-k+1),zα-ke-z+∑j=1kSjδ(k-j)(z), with 1⩽k⩽α. 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.