Algorithmic arithmetics with DD-finite functions

Abstract

Many special functions as well as generating functions of combinatorial sequences that arise in applications are D-finite, i.e., they satisfy a linear differential equation with polynomial coefficients. These functions have been studied for centuries and over the past decades various computer algebra methods have been developed and implemented for D-finite functions. Recently, we have extended this notion to DD-finite functions (functions satisfying linear differential equations with D-finite functions coefficients). Numerous identities for D-finite functions can be proven automatically using closure properties. These closure properties can be shown to hold for DD-finite functions as well. In this paper, we present the algorithmic aspect of these closure properties, discuss issues related to implementation and give several examples.