New opportunity for approximate calculation of dipole matrix elements between Rydberg atomic states

Asymptotic expansions of the Jacobi polynomials P-n((alpha,beta))(cos theta ) have been derived for the case when the parameters n, alpha, beta are lar ge and comparable in magnitude. The expansions have different forms in the middle of the interval 0 < theta < pi and near the limiting points theta = 0 and theta = pi. They turn into the previously known results if n much gre ater than alpha, beta, The obtained expressions could be used for analytic calculation and numerical computation of the atomic dipole matrix elements between Rydberg states in an extended range of quantum numbers.