Sp. Goreslavskii et Vp. Krainov, New opportunity for approximate calculation of dipole matrix elements between Rydberg atomic states, LASER PHYS, 9(6), 1999, pp. 1266-1269
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.