A uniform asymptotic expansion for Jacobi polynomials via uniform treatment of Darboux's method

Uniform asymptotic properties of the classical Jacobi polynomials have been studied via various approaches other than Darboux's method. In this note, by using ideas of the uniform treatment of Darboux's method, an asymptotic expansion, in terms of the Bessel function of the first kind and its derivative, is obtained for Jacobi polynomials . The expansion is uniformly valid for θ∈[0,π-ε], ε being an arbitrary positive constant.