Power Series Representations of Hypergeometric Type and Non-Holonomic Functions in Computer Algebra. Updated pdf file here.
I worked under the supervision of Prof. Dr. Wolfram Koepf. I give below brief details of my major results.
1. An algorithm, say mfoldHyper, that computes bases of the subspaces of all m-fold (m being a positive integer) hypergeometric term solutions of holonomic recurrence equations. This algorithm is new, and it has the advantage to linearize the computation of Laurent-Puiseux series: every linear combination of hypergeometric type series, even for many different values of m, is detected.
2. An algorithm that computes normal form representations for non-holonomic functions. A very important consequence of this algorithm is its ability to prove difficult identities.
3. A variant of van Hoeij's algorithm for computing hypergeometric term solutions of holonomic recurrence equations. Following the main steps of van Hoeij's algorithm, my variant computes hypergeometric term solutions of holonomic recurrence equations as fast as the original algorithm while using other methods. This algorithm is an integral part of mfoldHyper mentioned in 1.
Related papers can be found under the section Papers of the website.