Applied Mathematics and Computation 321 (2018) 121–129 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Ro-vibrational energies of the shifted Deng-Fan oscillator potential with Feynman path integral formalism H. Boukabcha a , M. Hachama b,∗ , A. Diaf a a Laboratory of Energy and Intelligent Systems, University of Khemis Miliana, Khemis Miliana 44225, Algeria b Laboratory of Pure and Applied Mathematics, University of M’Sila, Algeria a r t i c l e i n f o Keywords: Shifted Deng-Fan oscillator potential ℓ-states Feynman propagator a b s t r a c t In this paper, we derive approximate analytical solutions of the D-dimension Feynman propagator in presence of the shifted Deng-Fan oscillator by using an appropriate approx- imation of the centrifugal term. Energy eigenvalues and the corresponding eigenfunctions are obtained for different quantum numbers and some diatomic molecules. In the three- dimensional case, we calculate the energy level values and compare our results with those given in the literature. Our calculated energy levels are in much better agreement with the ones obtained numerically. © 2017 Elsevier Inc. All rights reserved. 1. Introduction Hyperbolic potentials are important molecular models used to describe the vibrations and the atoms interactions in poly- atomic molecules [1]. Recently, much efforts have been made to obtain analytical solutions of their ℓ-states in nonrelativistic and relativistic quantum mechanics. Among them, we can cite the Manning–Rosen [2,3], the Hulthén [4], the Rosen–Morse [5,6], the Eckart [7], and Schiöberg [8,9] potentials. Some of them are equivalent [10]. Furthermore, all these approaches share a basic idea of approximating the centrifugal term which makes the problem non-solvable. Such approximations write the centrifugal term in a hyperbolic form similar to the original potential. Some coefficients are then computed using a Taylor approximation of both sides. In this work, we investigate the shifted Deng-Fan oscillator [11] proposed back in 1957 for diatomic molecules. This potential, which attracted much interest lately, has the form V (r ) = D 1  1 − b e αr − 1  2 − D 2 , b = e αre − 1, (1) where D 1 is the dissociation energy, r e is the position of the minimum, and α denotes the radius of the potential. In the last decade, different techniques have been used to calculate arbitrary ℓ-states analytical solutions of the shifted Deng-Fan oscillator, based on the Schrödinger equation: the supersymetric shape invariance approach [12], the Nikiforov– Uvarov method [13], the asymptotic iteration technique [14], and the exact quantization rule approach [15]. Although progress have been achieved, there is still a need to develop new techniques to improve the approximations, especially for large values of n and ℓ and different values of α. ∗ Corresponding author. E-mail address: hachamam@gmail.com (M. Hachama). https://doi.org/10.1016/j.amc.2017.10.044 0096-3003/© 2017 Elsevier Inc. All rights reserved.