Mahler systems, differential equations, and their solutions

Mathematical models in natural science typically are described in terms of equations that relate the quantities involved. A significant part of mathematics research deals with investigating properties and solution methods of such equations. Various types of equations can be distinguished according to their structural properties. For example, differential equations relate values of functions and their derivatives at the same place, and recurrence equations relate values of functions at several places with fixed offset. Both are types of functional equations. In the project, we will develop algorithms for solving and constructing selected types of linear functional equations. Regarding solutions, we will exclusively be interested in explicit expressions for exact solutions rather than computing approximate solutions. For linear differential or recurrence equations, many algorithms for finding explicit solutions of different kinds are known. Even algorithms that directly solve systems of such equations have been developed. For so-called Mahler equations, however, the development of solution methods has started in recent years only and is not as advanced yet. Mahler equations relate values of functions at places determined by powering. They have been named after Kurt Mahler, who used them to study transcendence of numbers almost 100 years ago. One goal of the project is to develop efficient algorithms that directly solve linear systems of Mahler equations. Another goal of the project is to further develop and refine algorithms for solving linear differential equations allowing special functions appearing in the equations as well as in the solutions. Also the converse, i.e. finding for a given function an equation that is satisfied by it, is relevant in certain situations, especially when the function is not given in a convenient explicit form. For example, if a function is given by an integral or infinite sum, then an equation satisfied by that function can allow to determine further properties (e.g. singularities) or even an explicit form more easily. There are algorithms to compute such equations and often the resulting equations can then be decomposed into simpler equations that are easier to deal with than the original equation. A goal of the project is to develop algorithms that are able to directly compute such decomposition into simpler equations by exploiting the internal structure of the given integral or sum, without first computing the larger equation.