On a Separation Theorem for the Zeros of the Ultraspherical Polynomials

1. It will be recalled that the ultraspherical polynomials are those which are orthogonal on the interval ( — 1, 1), corresponding to the weight function (1— x2)x~1/2, X>—1/2. In what follows X = 0 will also be excluded. The coefficients of these polynomials are functions of the parameter X appearing in the weight function, and the symbol P„(x, X), indicative of this fact, will be used to denote the ultraspherical polynomial of degree ». This paper is mainly concerned with a separation theorem for the zeros of the ultraspherical polynomials and the zeros of the polynomials derived from the ultraspherical polynomials by differentiation with respect to the parameter X, under various conditions of normalization. In addition some generalizations to other related polynomials are obtained, and an application is made also to a limited class of the Jacobi polynomials, which include the ultraspherical polynomials as special cases. To facilitate the exposition certain relations are listed below for reference. These are taken from Orthogonal polynomials, Gabor Szegö, Amer. Math. Soc. Colloquium Publications, vol. 23, New York, 1939. The first number in the reference bracket will denote the page, and the second number the relation, in the work just mentioned.