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In this paper, we investigate a discrete relativistic Toda lattice ( dRTL + (α) ) system, which may describe particle vibrations in lattices with an exponential interaction force. First, we construct its discrete generalized (m , 2 N − m) -fold Darboux transformation, from which we can explicitly give its analytic solutions, such as discrete multi-soliton solutions, position controllable rational and semi-rational solutions and their hyperbolic-and-rational mixed solutions, whose properties and dynamics are analyzed and shown graphically. Second, the asymptotic behaviors of diverse exact solutions are analyzed, which shows that the interactions among different solutions are always elastic. In particular, the position of controllable rational solutions and asymptotic state analysis of discrete hyperbolic-and-rational mixed solutions are obtained and discussed for the first time. Finally, we study some integrable properties of this system, such as the integrable hierarchy and relevant Hamiltonian structures and conservation laws from a discrete spectral problem. These results may be helpful for understanding nonlinear lattice dynamics. [ABSTRACT FROM AUTHOR]