# F50-11: Certificate-Free Summation and Integration

Combinatorial identities, integrals involving special functions, and other summation and integration formulas are nowadays frequently evaluated resp. proved by computer algebra methods that are based on recurrence equations or differential equations. The method of choice to produce such equations is called creative telescoping. It suffers from the fact that most available algorithms compute, in addition to the telescoper, i.e., the operator representing the sought equation, a so-called certificate. The latter, however, is not needed in many situations. The goal of the proposed project is to design algorithms that avoid the computation of the certificate and therefore are expected to be much more efficient than existing algorithms. In special cases, such as rational functions, hypergeometric terms, hyperexponential functions, this hope was already fulfilled. We aim at generalizing these results to the larger class of holonomic functions and to difference and differential fields, yielding valuable tools for solving problems from combinatorics and other areas of mathematics. By means of the holonomic ansatz, we also obtain a powerful machinery for evaluating symbolic determinants, which arise frequently in combinatorial problems.