On the approximation of the Jacobi polynomials

New approximations of the Jacobi polynomials P-n((alpha,beta)) (x) are provided on the interval (1, infinity). The approximations are given explicitly in terms of some expressions derived from a coefficient of a related hypergeometric equation and in terms of certain perturbation terms. The perturbation terms are essentially resolvent series that are absolutely convergent. These series converge uniformly for all positive n, alpha and beta in some semi-infinite interval and for x in the interval [1, infinity). They are shown to converge faster then a geometric series, where the ratio of successive terms is pi(2) /32. We thus also demonstrate that it is possible to approximate the Jacobi polynomials in the vicinity of x = 1 as well as on the entire interval [1, infinity) without resorting to Bessel functions. The asymptotic approximation of P-n((alpha + alpha n, beta + bn)) (x) on [1, infinity), that is related to the Racah coefficients and to a one-dimensional quantum walk, is also pointed out.