Vol. 72 (2013) REPORTS ON MATHEMATICAL PHYSICS No. 1 MEIJER’S G-FUNCTIONS SERIES AS EXACT SOLUTIONS OF THE RADIAL LINEAR POTENTIAL M. A. BENBITOUR and M. T. MEFTAH Physics Department, LRPPS Laboratory UKMO, Ouargla 30 000, Algeria (e-mails: mohamedbenbitour@yahoo.fr, mewalid@yahoo.com) (Received March 11, 2013 – Revised May 22, 2013) We present an analytic solution to the Schr¨ odinger equation for a particle in a radial linear potential and for arbitrary value of orbital momentum. The solution is given in terms of a series sum of Meijer’s functions. We derive the solution by solving it iteratively as an alternative approach to the technics of combinatorics functions. The energy eigenvalues are solutions of a trenscendental equation involving Meijer functions. Keywords: linear potential, Schr¨ odinger equation, Meijer’s function, Mellin transform. 1. Introduction Quark-antiquark bound states are explained by several potential models like the Martin potential [1], Cornell potential [2] and Richardson potential [3]. These potentials include the short distance Coulomb interaction of quarks, and the large distance quark confinement by the linear term potential. It seems that the linear potential plays the same role in particle physics as the Coulomb potential in atomic physics. There are no solutions in terms of usual functions for Schr¨ odinger equation with a spherical linear potential. There is no transformation of the associated radial Schr¨ odinger equation that makes possible to bring it into the hypergeometric form. So there are no solutions in terms of special functions. The only used methods are the power series solutions that lead to a three-term recursion relations with nonconstant coefficients. Long ago A. F. Antippa and A. J. Phares, authors of [8], introduced a general formalism, using combinatorics functions, for solving linear multiterm recursion relations with nonconstant coefficients. This formalism was subsequently applied to solve the Schr¨ odinger equation with a linear potential. A. F. Antippa et al. obtained the energy eigenvalues in [12], and in [13] completed the study of the wave functions. Much more recently, the authors of [7] made further progress in the field by providing the wave functions for the combined linear and Coulomb potentials using the combinatorics function technique. The wave function is obtained as a series power expansion, where the coefficients are given in terms of functionals, called structure functions. The energy eigenvalues were given by the roots of an [93]