Abstract
Two sets with an infinite number of new systems of orthogonal polynomials have recently been discovered by Smith (1982) in connection with some nonlinear physical problems, e.g. the dispersion of a buoyant contaminant in a fluid. They appear as solutions of nonlinear differential equations. Let (Pn(x; m, k, delta )) and (Qn(x; m,k)), with n=0, k, m+k, 2m, 2m+k,. . ., denote a generic system of each set. The positive integers k and m are restricted by k(m and delta )1-k. Although the orthogonality interval of these polynomials is real, their zeros are generally complex. Here the sum rules yr= Sigma xri,n, r=1,2,. . ., for the zeros (xi,n; i=1,2,. . ., n) of the n-th degree polynomials of these sets are studied. It is found that all these quantities vanish except for r=pm, p being an arbitrary positive integer. Simple recurrent expressions for ypm are given.