A sequence is called C-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with C-finite coefficients. Recently, it was shown that such C2-finite sequences satisfy similar closure properties as C-finite sequences. In particular, they form a difference ring.

A sequence is called C-finite, if it satisfies a linear recurrence with constant coefficients and holonomic, if it satisfies a linear recurrence with polynomial coefficients. The class of C2-finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients whose leading coefficient has no zero terms. Recently, we investigated computational properties of $C^2$-finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring.

Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients and also referred to as D-finite sequences. A subclass are C-finite sequences satisfying a linear recurrence with constant coefficients.

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.

The most common techniques for obtaining numerical solutions to partial differential equations on non-trivial domains are (high order) finite element methods. The given domain is subdivided into simple geometric objects and an approximate solution is computed as a linear combination of locally supported basis functions. In the past decade there have been several successful collaborations between mathematicians from numerical analysis and computer algebra to analyze or improve numerical methods. The applied methods include Gröbner bases, Cylindrical Algebraic Decomposition, algorithms for special functions, etc. In this tutorial we plan to present an introduction to the basic concepts of finite element methods and we want to conclude with an overview on some of those recent collaborations and the involved proof techniques.

Proving positivity of a sequence given by a linear recurrence with polynomial coefficients (P-finite recurrence) is a non-trivial task for both humans and computers. Algorithms dealing with this task are rare or non-existent. One method that was introduced in the last decade by Gerhold and Kauers succeeds on many examples, but termination of this procedure has been proven so far only up to order three for special cases. Here we present an analysis that extends the previously known termination results on recurrences of order three, and also provides termination conditions for recurrences of higher order.

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed on a great many examples, there is no guarantee for them to terminate, and they do in fact not terminate for every input. For some restricted classes of P-finite recurrence equations of order up to three we provide a priori criteria that assert the termination of the algorithms.