Uniform asymptotic expansion of the Jacobi polynomials in a complex domain

Abstract

An asymptotic formula is found that links the behaviour of the Jacobi polynomial Pnα,β)(z) in the interval of orthogonality [–1,1] with that outside the interval. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre.