Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials

For each m (= 1, . . . , n) the nth Laguerre polynomial L (α) n (x) has an m-fold zero at the origin when α =−m. As the real variable α→−m, it has m simple complex zeros which approach 0 in a symmetric way. This symmetry leads to a finite value for the limit of the sum of the reciprocals of these zeros. There is a similar property for the zeros of the q-Laguerre polynomials and of the Jacobi polynomials and similar results hold for sums of other negative integer powers. © 2007 Elsevier Inc. All rights reserved.