In atomic or molecular physics it is customary the use of sum rules to handle the matrix elements that appear in many computations. One of the conspicuous example is the Blanchard relation [1] which is a useful recurrence formula for non-diagonal, non-relativistic, arbitrary matrix elements. However, this often involves matrix elements corresponding to parity-forbidden transitions, which have to be eliminated since they are physically irrelevant. A more general expression for the Coulomb bound-free irregular electric multipoles has been given in terms of the Apple F₂ function. For multipolar order beyond the dipole, these functions generally do not reduce to simpler hypergeometric functions.In the intervening years a simplified method [2] has been proposed for non-relativistic case. These formulae are particularly well suited for the study of autoionization of large-angular momentum doubly excited Rydberg states. The recursion formulae are no more mathematical than special cases of recursions on Apple functions but are quite powerful with respect to numerical accuracy and extremely efficient with respect to computation speed.In this work we present preliminary results on the main properties of non-relativistic Coulomb matrix elements between circular Rydberg states and continuum in Al X ion . Our computation method is based on the model suggested in Ref.2. The atomic system is described by a closed-shell core and two active electrons, named valence and Rydberg electron. Then, in a single configuration description, the Rydberg electron is assumed to evolve from a bound hydrogenic state to a continuum state, while the valence electron falls from the excited ionic state to a lower state. The electronic repulsion 1/r₁₂ is generally stronger than spin-orbit effects for the Rydberg electron. In Biedenharn’s notation, the matrix element for bound-free transition is:
(λ,q;l)≡∫₀^∞drr^−qF_ηlP_nl+λ
where F_ηl is a continuum wavefunction of energy 1/2η² and angular momentum l and where P_nl+λ is a discrete wavefunction of energy –1/2n² and angular momentum l + λ. The basic recursion relation contains only radial integrals with q-λ of a given parity. This relation allows the multipole (λ-1,q+1;l+1) to be expressed using multipoles of the two preceding orders.
Free-bound matrix elements are calculated for <ηl+2/r^−q/nl> for different l,η,n .

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