Elimination in Operator Algebras

A large class of special functions are solutions of systems of linear difference and differential equations with polynomial coefficients. For a given function, these equations considered as operator polynomials generate a left ideal in a noncommutative algebra called Ore algebra. This ideal with finitely many conditions characterizes the function uniquely so that Gröbner basis techniques can be applied. Many problems related to special functions which can be described by such ideals can be solved by performing elimination of appropriate noncommutative variables in these ideals. In this work, we mainly achieve the following:

1. We give an overview of the theoretical algebraic background as well as the algorithmic aspects of different methods using noncommutative Gröbner elimination techniques in Ore algebras in order to solve problems related to special functions.
2. We describe in detail algorithms which are based on Gröbner elimination techniques and perform the creative telescoping method for sums and integrals of special functions.
3. We investigate and compare these algorithms by illustrative examples which are performed by the computer algebra system Maple. This investigation has the objective to test how far noncommutative Gröbner elimination techniques may be efficiently applied to perform creative telescoping.