Evaluation of matrix elements of rk and recurrence relations for the Klein–Gordon equation with a Coulomb field

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We obtain two useful recurrence relations between diagonal matrix elements for the Klein–Gordon equation with a Coulomb potential. Some explicit expressions of diagonal matrix elements of rk(6 ⩾ k ⩾ −6) are presented. The general calculation formulae for off-diagonal matrix elements ⟨n′1l′1|rk|n′2l′2⟩ are derived analytically. An important five-term recurrence relation among off-diagonal matrix elements, the analogue to what is known as Blanchard's rule in the non-relativistic Schrödinger equation case, is obtained explicitly.