A characterization of polynomial time computable functions from the integers to the reals using discrete ordinary differential equations

In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, we pointed out the fundamental role of linear (discrete) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming. In this article, we extend the approach to a characterization of functions from the integers to the reals computable in polynomial time in the sense of computable analysis. In particular, we provide a characterization of such functions in terms of the smallest class of functions that contains some basic functions, and that is closed by composition, linear length ODEs, and a natural effective limit schema.