Evaluation of matrix elements of r^k and recurrence relations for the Klein-Gordon equation with a Coulomb field

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 r^k(6 ≥ k ≥ -6) are presented. The general calculation formulae for off-diagonal matrix elements 〈n′₁l′₁|r^k|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.