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