A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error Bounds

Abstract

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [ 13 , p. 197]: (1.1) where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ 1/2 O ( n –3/2 ).