Representation of a solution to a class of linear discrete Langevin equations

This paper introduced, for the first time, a novel discrete Langevin equation as a new generalization of the classical Langevin model, involving two distinct integer-order differences. A new special function generated by the system parameters was defined to obtain explicit analytical solutions of the associated initial value problem. Several special cases were discussed, and open problems were proposed. The results established a new theoretical framework for discrete Langevin-type systems and contributed to the qualitative theory of difference equations.