D-dimensional Klein–Gordon equation in presence of deformed generalized Deng–Fan with Yukawa potential class: Approximate bound state solutions in relativistic and non-relativistic regimes

In this paper, we examine the bound state solutions of the $D$-dimensional Klein–Gordon equation for a deformed Deng–Fan potential in generalized form along with Yukawa potential class. The supersymmetric quantum mechanics method and Nikiforov–Uvarov method are employed, utilizing a proper correspondence to the centrifugal potential term. Remarkably, in both methods, we obtain the same analytical expressions for normalized wave functions and energy eigenvalues, expressed in closed form by hypergeometric functions and Jacobi polynomials, for all $l$ and $n$ quantum states. Additionally, we explore the thermodynamic quantities (internal energy, partition function, specific heat, entropy and free energy) in case of both relativistic and non-relativistic regimes for the aforementioned potential. Finally, we demonstrate energy variation against various potential parameters for a few diatomic molecules pictorially.