Integral D-finite Functions

Integers are one of the foundations of mathematics. They can be added and multiplied but not always divided. In order to enable also division by all numbers (except zero), we can switch to the larger domain of rational numbers. Besides the usual integers and rational numbers, other number systems are considered in number theory, for example so-called algebraic numbers. Among the algebraic numbers we can distinguish some that are more similar to integers from other that are more similar to rational numbers. Those of the first kind are called algebraic integers. The concept of algebraic integers plays a crucial role in classical number theory. The notion of integrality can be extended from numbers to functions. For so-called algebraic functions it leads to a theory which is very similar to the theory of algebraic numbers. In the project we study a generalization of the notion of integrality to a larger class of functions, so-called D-finite functions. These functions play an important role in many areas of pure and applied mathematics, for two reasons: first, because they naturally arise in many different contexts, and secondly because it is easy to do computations with them. In particular, properties of D-finite functions can often be found and proved automatically using suitable computer programs. Computer programs for computing with D-finite functions have been subject to research for many years. The notion of integrality for D-finite functions, which has only been introduced recently, has so far not played a great role in these developments. But in view of the usefullnes of this notion in number theory and in the theory of algebraic functions it is clear that this concept also has a great potential for D-finite functions. It is the goal of this project to exploit this potential. To do so, we will both clarify theoretical aspects of integral D-finite functions and develop applications to computer algebra. We expect that by this work some classical problems can be solved more efficiently and that for some new kinds of problems we can formulate algorithmic solutions for the first time. The investigation of integral D-finite functions is therefore not only of immediate theoretical interest but is indirectly also important to those scientific areas in which questions about D-finite functions arise which can be answered using computer algebra. These include in particular combinatorics and experimental mathematics but also certain areas of applied mathematics.

Integers are one of the foundations of mathematics. They can be added and multiplied but not always divided. In order to enable also division by all numbers (except zero), we can switch to the larger domain of rational numbers. Besides the usual integers and rational numbers, other number systems are considered in number theory, for example so-called algebraic numbers. Among the algebraic numbers we can distinguish some that are more similar to integers from other that are more similar to rational numbers. Those of the first kind are called algebraic integers. The concept of algebraic integers plays a crucial role in classical number theory. The notion of integrality can be extended from numbers to functions. For so-called algebraic functions it leads to a theory which is very similar to the theory of algebraic numbers. In the project we studied a generalization of the notion of integrality to a larger class of functions, so-called D-finite functions. These functions play an important role in many areas of pure and applied mathematics, for two reasons: first, because they naturally arise in many different contexts, and secondly because it is easy to do computations with them. In particular, properties of D-finite functions can often be found and proved automatically using suitable computer programs. Computer programs for computing with D-finite functions have been subject to research for many years. The notion of integrality for D-finite functions, which has only been introduced recently, has so far not played a great role in these developments. But in view of the usefulness of this notion in number theory and in the theory of algebraic functions it is clear that this concept also has a great potential for D-finite functions. It is the goal of this project was exploit this potential. To do so, we have clarified theoretical aspects of integral D-finite functions and developed applications to computer algebra. This work has contributed for example to the development of reduction-based creative telescoping algorithms for D-finite functions and for solving differential equations. These investigations of integral D-finite functions are not only of immediate theoretical interest but are indirectly also important to those scientific areas in which questions about D-finite functions arise which can be answered using computer algebra. These include in particular combinatorics and experimental mathematics but also certain areas of applied mathematics.