Inequalities for zeros of associated polynomials and derivatives of orthogonal polynomials

Abstract

It is well known and easy to see that the zeros of both the associated polynomial and the derivative of an orthogonal polynomial p n(x) interlace with the zeros of p n(x) itself. The natural question of how these zeros interlace is under discussion. We give a sufficient condition for the mutual location of kth, 1⩽k⩽n−1, zeros of the associated polynomial and the derivative of an orthogonal polynomial in terms of inequalities for the corresponding Cotes numbers. Applications to the zeros of the associated polynomials and the derivatives of the classical orthogonal polynomials are provided. Various inequalities for zeros of higher order associated polynomials and higher order derivatives of orthogonal polynomials are proved. The results involve both classical and discrete orthogonal polynomials, where, in the discrete case, the differential operator is substituted by the difference operator.