Alternation points, weights and orthogonal polynomials

Abstract

In this note we characterize the orthogonal polynomials having alternation points in terms of algebraic properties of the coefficients of their recurrence relations. As an example, we deduce that the four types of Chebyshev polynomials are the only Jacobi polynomials of degree greater than 3 that have alternation points. We also show that if there are alternation points for polynomials orthogonal with respect to a measure μ , then the inner product defined by μ is a multiple of the discrete inner product generated by the alternation points. Finally, we produce orthonormal polynomials having alternation points that are the roots of a given trigonometric equation.