INTEGRAL REPRESENTATIONS FOR JACOBI POLYNOMIALS AND SOME APPLICATIONS

An integral for [Pn(α + μ,β)(x)] [Pn(α + μ,β)(1)] in terms of [Pn(α,β)(y)] [Pn(α,β)(1)] with a positive kernel is obtained. For β = ± 1 2 this integral is equivalent to an important integral of Feldheim and Vilenkin connecting ultraspherical polynomials. As an application we show that Pn(α,α)(x) Pn(α,α)(1) = ∫-11 Pn(β,β)(y) Pn(β,β)(1) dμ(y) where α > β ≥ - 1 2, - 1 ≤ x ≤ 1, and dμ(y) is a positive measure which depends on x but not n. For β = - 1 2 this is a result of Seidel and Szasz. Similar results are obtained for Jacobi polynomials and the positivity of certain sums of ultraspherical and Jacobi polynomials is obtained. © 1969.