A recurrence formula for obtaining certain matrix elements in the base of eigenfunctions of the Hamiltonian for a particular screened potential

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Abstract
A recurrence formula for obtaining matrix elements of products of powers of exp(- nu r) and 1-exp(- nu r) is derived. The matrix elements are calculated in the base of eigenfunctions of the Hamiltonian for the effective potential - lambda exp(- nu r)/(1-exp( nu r))+ mu exp( nu r)/(1-exp( nu r))2, where lambda , nu and mu are positive constants. This potential can be considered as a generalisation of a potential suggested by Hylleraas and Risberg (1941) and by Hulthen (1942). In the limit of the parameter nu tending to zero the recurrence formula is transformed into a recurrence formula is transformed into a recurrence formula given by Badawi et al. (1973) for matrix elements of powers of r for the hydrogen atom.