Relative Extrema, Zeros, and Inequalities for Jacobi Functions of the First andSecond Kind

摘要

Let P sup alpha, beta sub nu (x) denote the Jacobi function of the first kind. AJacobi function of the second kind, Q sub alpha, beta sub nu (x) is introduced, in analogy with the ultraspherical or Gegenbauer function of the second kind, D sup lambda sub nu (x), introduced by Durand. Inequalities for these functions and for their relative extrema on -1 less than x less than 1, generalizing thereby similar facts known for the Jacobi polynomials P sup alpha, beta sub n (x), n = 0,1, ..., with alpha less than beta = -1/2, and the P and Q functions above.