On two sets of orthogonal polynomial systems encountered in nonlinear physics

Abstract

Two sets with an infinite number of new systems of orthogonal polynomials have recently been discovered by Smith in connection with some non-linear physical problems, e.g. the dispersion of a buoyant contaminant in a fluid. They appear as solutions of non-linear differential equations. Let {P,(x; m, k, S)} and (Q,(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 1 - k. Although the orthogonality interval of these polynomials is real, their zeros are generally complex. Here the sum rules y, = X x;., r = 1.2,. . ., for the zeros {x ,,,,; i = 1,2,. . . , n} of the nth-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 ypn, are given.