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.
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