Widened derangements and generalized Laguerre polynomials

Let Dh, Ek and Falpha be sets of size h; k; alpha respectively, with k less than or equal to h. A permutation of the union of Dh, Ek and Falpha such that the elements of Dh are not fixed and the elements of Ek cannot occupy a site originally occupied by an object of the same type or by an object of Falpha will be called a strongly widened derangement. We show a connection between strongly widened derangements and generalized Laguerre polynomials that provides a generalization, for integer values of alpha, of Even and Gillis (1976) different from the one presented in Foata and Zeilberger (1988)..