The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator identities. In the present paper, we extend and adapt the framework to the context of rewriting and polynomial reduction. We give a sufficient condition on the polynomials used for rewriting to ensure that standard polynomial reduction automatically respects domains and codomains of operators. Finally, we adapt the noncommutative Buchberger procedure to compute additional compatible polynomials for rewriting. In the package OperatorGB, we also provide an implementation of the concepts developed.

We propose a general algorithmic approach to noncommutative operator algebras generated by linear operators. Ore algebras are a well-established tool covering many cases arising in applications. However, integro-differential operators, for example, do not fit this structure. Instead of using (parametrized) Gröbner bases in noncommutative polynomial algebras as has been used so far in the literature, we use Bergman's basis-free analog in tensor algebras. This allows for a finite reduction system with unique normal forms. To have a smaller reduction system, we develop a generalization of Bergman's setting, which also makes the algorithmic verification of the confluence criterion more efficient. We provide an implementation in Mathematica and we illustrate both versions of the tensor setting using integro-differential operators as an example.

Integrals are important to many applications ranging from physics over engineering to statistics. While systematic methods for symbolic computation of integrals have a long history, computer algebra tools for computation of parameter integrals are a more recent topic. For computing (definite) parameter integrals, one does not necessarily need to know an explicit antiderivative of the integrand, such computations often rely on techniques like differentiation under the integral sign instead.

An algorithm for parametric elementary integration over differential fields constructed by a differentially transcendental extension is given. It extends current versions of Risch's algorithm to this setting and is based on some first ideas of Graham H. Campbell transferring his method to more formal grounds and making it parametric, which allows to compute relations among definite integrals. Apart from differentially transcendental functions, such as the gamma function or the zeta function, also unspecified functions and certain families of iterated integrals such as the polylogarithms can be modeled in such differential fields.

Let F be a differential field generated from the rational functions over some constant field by one hyperexponential extension. We present an algorithm to compute the solutions in Fn of systems of n first-order linear ODEs. Solutions in F of a scalar ODE of higher order can be determined by an algorithm of Bronstein and Fredet. Our approach avoids reduction to the scalar case. We also give examples to show how this can be applied to integration.