This work presents the concept of weighted pseudo S-asymptotically $(N,\lambda )$ -periodic sequences, which can be seen as a generalization of discrete weighted pseudo S-asymptotically λ-periodic and λ-antiperiodic sequences. We prove fundamental properties of such sequences such as completeness of the space, convolution and superposition theorems. These results lay the groundwork for investigating the existence and uniqueness of weighted pseudo S-asymptotically $(N,\lambda )$ -periodic solutions to a specific class of abstract semilinear difference equations of convolution type, where the scalar kernels are given in terms of the Poisson transformation. Here we recall important properties of Poisson transformation and prove that it maps completely monotone functions in completely monotone sequences, maintaining the crucial feature of complete monotonicity across different domains. We present constructive examples to showcase the feasibility of the stated hypotheses, along with two numerical simulations that provide insight into the behavior of these functions as solutions to fractional-order difference equations.
