Arbitrary $$\ell$$-state solutions of the Klein–Gordon equation with the Eckart plus a class of Yukawa potential and its non-relativistic thermal properties

We report bound state solutions of the Klein Gordon equation with a novel combined potential, the Eckart plus a class of Yukawa potential, by means of the parametric Nikiforov-Uvarov method. To deal the centrifugal and the coulombic behavior terms, we apply the Greene-Aldrich approximation scheme. We present any $\ell$-state energy eigenvalues and the corresponding normalized wave functions of a mentioned system in a closed form. We discuss various special cases related to our considered potential which are utility for other physical systems and show that these are consistent with previous reports in literature. Moreover, we calculate the non-relativistic thermodynamic quantities (partition function, mean energy, free energy, specific heat and entropy) for the potential model in question, and investigate them for a few diatomic molecules. We find that the energy eigenvalues are sensitive with regard to the quantum numbers $n_r$ and $\ell$ as well as the parameter $\delta$. Our results show that energy eigenvalues are more bounded at either smaller quantum number $\ell$ or smaller parameter $\delta$.