### Symbolic Summation In Differential Fields

We seek to understand a general method for finding a closed form for a given sum that acts as
it’s ``antidifference`` in the same way that an integral has an antiderivative. Once an antidifference is found,
then given the limits of the sum, it suffices to evaluate the antidifference at the given limits. When the summand
is hypergeometric, Gosper’s algorithm finds the antidifference, if it exists.

Many frequently encountered
sequences however are not hypergeometric, such as the harmonic numbers and their generalizations. In this case
the summation is done by Karr's algorithm. The algorithm finds antidifferences by redefining the problem
in terms of difference fields (a field equipped with an automorphism.)
Once translated into a difference field
setting, Karr’s algorithm finds a solution to the corresponding difference equation, and if no solution
is found, adjoins a symbolic solution which represents the sequence we are trying to sum up. Carsten Schneider
expanded Karr’s method of solving first order difference equations to higher
order equations in special cases.

Karr’s original papers leave out several of the key proofs needed to implement
the algorithm.
We attempt fill in these gaps by solving two major problems: the equivalence problem and the homogenous group
membership problem. Solving these two problems
is essential to finding the polynomial degree bounds and denominator bounds for solutions of difference
equations.

**PAPERS AND PRESENTATIONS:**

- Senior Thesis by Javier Vazquez-Trejo, SCS, CMU, 2014.

*Last updated Wednesday, May 07, 2014*