Computation of Newton sum rules for polynomial solutions of O. D. E. with polynomial coefficients (Q2726117)

In this paper the authors study the distribution of zeros of polynomial solutions \(P_N\) of the differential equation NEWLINE\[NEWLINE \sum_{i=0}^m g_i(x) f^{(i)}(x)=0, NEWLINE\]NEWLINE where \(g_i\) are polynomials of degree \(c_i\). In fact they are interested to compute the so called Newton sum rules defined by NEWLINE\[NEWLINE y_k=\sum_{i=1}^N x_i^k,\qquad x_i\quad\text{zero of \(P_N\)}. NEWLINE\]NEWLINE For doing that, the authors use the results by \textit{E. Buendía, S. Dehesa} and \textit{F. J. Gálvez} [In: Orthogonal Polynomials and their Applications. Lecture Notes Math. 1329, 222-235 (1988; Zbl 0643.34023)] consisting on a recursive formula for finding the quantities \(y_k\) in terms of the so-called Case sum rules. Their main contribution in this paper are contained in two propositions. The first one states that the Case sum rules can be written in terms of the so-called generalized Lucas polynomials of second kind, and the second one states that the Newton sum rules \(y_k\) can be computed by means of the generalized Lucas polynomials of first kind. This allows to obtain a numerical algorithm for computing the Newton sum rules for several families of polynomials that is demonstrated in an example at the end of the paper.