PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 83, Number 4, December 1981 A MONOTONIC PROPERTY FOR THE ZEROS OF ULTRASPHERICAL POLYNOMIALS ANDREA LAFORGIA1 Abstract. It is shown that Xx¡$ increases as X increases for 0 < X < 1, k = I, 2,..., [|J where xfy is the kth positive zero of ultraspherical polynomial P?\x). The aim of this work is to prove the following Theorem. Let xjfy, k = 1, 2, . . ., [f ], be the zeros of the ultraspherical polynomial F„(X)(x) in decreasing order on (0, 1), where 0 <X < 1. Then for every e > 0, XxfX < (X + e)x£r\ *=l,2,...,[f]. Remark 1. For our purposes the following form of Sturm comparison theorem will prove useful. This formulation differs from the usual formulation [2, p. 19] in that fix) < F(x) is hypothesized for the interval a < x < Xm, rather than for the larger interval a < x < xm. (See the work [1] for the proof of this formulation of Sturm theorem.) Lemma. Let the functions y(x) and Y(x) be nontrivial solutions of the differential equations y"(x) + fix)y(x) = 0; Y"(x) + F(x) Y(x) = 0 and let them have consecutive zeros at xx, x2, .. . , xm and Xx, X2, . . . , Xm respec- tively on an interval (a, b). Suppose that fix) and F(x) are continuous, that fix) < F(x), a < x < Xm, and that (0 lim+ [y'(x)Y(x) - y(x)Y'(x)] = 0. x—>cr -1 Then Xk < xk, k = 1, 2, . . . , m. Proof of the theorem. The function «(x) = (1 - x2)A/2+1/4P^(x) Received by the editors March 6, 1981. 1980 Mathematics Subject Classification. Primary 33A45; Secondary 34A50. Key words and phrases. Zeros of ultraspherical polynomials, Sturm comparison theorem. 'This research was supported by Consiglio Nazionale delle Ricerche. © 1981 American Mathematical Society 0002-9939/81/0000-0568/01.50 757 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use