RELATIVISTIC SYMMETRIES OF BOSONIC PARTICLES AND ANTIPARTICLES IN THE BACKGROUND OF THE POSITION-DEPENDENT MASS FOR THE IMPROVED DEFORMED HULTH?N PLUS DEFORMED TYPE-HYPERBOLIC POTENTIAL IN 3D-EQM SYMMETRIES

The bound state solutions of the deformed Klien-Gordon equation have been determined in the three-dimensional extended relativistic quantum mechanics 3D-ERQM symmetries using position-dependent mass (PDM) with unequal scalar and vector potential for the improved Hulthen plus improved deformed type-hyperbolic potential (PDM-SVID(H-TP)) models. PDM with unequal scalar and vector potential for the Hulthen plus deformed type-hyperbolic potential (PDM-(SVH-DTP)) models, as well as a combination of radial terms, which are coupled with the coupling LO , which explains the interaction of the physical features of the system with the topological deformations of space-space. The new relativistic energy eigenvalues have been derived using the parametric Bopp shift method and standard perturbation theory which is sensitive to the atomic quantum numbers ( j, l, s, m ), mixed potential depths (V0, S0, V1, S1), the rest, and perturbed mass (m0,m1 ), the screening parameter's inverse a , and noncommutativity parameters (O, r,z). Within the framework of 3D-ERQM symmetries, we have treated certain significant particular instances that we hope will be valuable to the specialized researcher. We have also treated the nonrelativistic limit and applied our obtained results to generate the mass spectra of heavy-light mesons (HLM) such as cc and bb under PDM-SE with improved deformed Hulthen plus improved hyperbolic potential (PDM-ID(H-TP)) models. When the three simultaneous limits (O,r,z) were applied, we recovered the normal results of relativistic in the literature (0,0,0 ) for the PDM-ID(H-TP)) models.