Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

In this paper, we give a procedure for discretizing recursion operators by utilizing unified bilinear forms within integrable hierarchies. To illustrate this approach, we present unified bilinear forms for both the AKNS hierarchy and the KdV hierarchy, derived from their respective recursion operators. Leveraging the inherent connection between soliton equations and their auto-B\"acklund transformations, we discretize the bilinear integrable hierarchies and derive discrete recursion operators. These discrete recursion operators exhibit convergence towards the original continuous forms when subjected to a standard limiting process.