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The relativistic wave equations determine the dynamics of quantum fields inthe context of quantum field theory. One of the conventional tools for dealingwith the relativistic bound-state problem is the Klein-Fock-Gordon equation. Inthis work, using a developed scheme, we present how to surmount the centrifugalpart and solve the modified Klein-Fock-Gordon equation for the linearcombination of Hulth\'en and Yukawa potentials. In particular, we show that therelativistic energy eigenvalues and corresponding radial wave functions areobtained from supersymmetric quantum mechanics by applying the shape invarianceconcept. Here, both scalar potential conditions, which are whether equal andnon-equal to vector potential, are considered in the calculation. The energylevels and corresponding normalized eigenfunctions are represented as arecursion relation regarding the Jacobi polynomials for arbitrary $l$ states.Beyond that, a closed-form of the normalization constant of the wave functionsis found. Furthermore, we state that the energy eigenvalues are quite sensitivewith potential parameters for the quantum states. The non-relativistic andrelativistic results obtained within SUSY QM overlap entirely with the resultsobtained by ordinary quantum mechanics, and it displays that the mathematicalimplementation of SUSY quantum mechanics is quite perfect.

Analytical bound-state solutions of the Klein-Fock-Gordon equation for the sum of Hulth\'en and Yukawa potential within SUSY quantum mechanics