Solution of a generalized Stieltjes problem

We present the exact solution for a set of nonlinear algebraic equations 1/ zl = pid + 2d/n Sigma (m not equall) 1/z(l)-z(m). These were encountered by us in a recent study of the low-energy spectrum of the Heisenberg ferromag netic chain. These equations are low-d (density) 'degenerations' of a more complicated transcendental equation of Bethe's ansatz for a ferromagnet, bu t are interesting in themselves. They generalize, through a single paramete r, the equations of Stieltjes, x(l) = Sigma (m not equall) 1/(x(l) - x(m)), familiar from random matrix theory. It is shown that the solutions of thes e set of equations are given by the zeros of generalized associated Laguerr e polynomials. These zeros are interesting, since they provide one of the f ew known cases where the location is along a nontrivial curve in the comple x plane that is determined in this work. Using a 'Green function' and a sad dle point technique we determine the asymptotic distribution of zeros.