Applications of computer algebra to the theory of hypergeometric series

This thesis consists of two parts which both deal with the application of computers to the theory of hypergeometric series. In the first part of this thesis we study how symbolic computational software, like Maple, can be used to generate hypergeometric transformations systematically. Based on the observation that ∑i(∑ j aij) = ∑j(∑ i aij), we generate double sums which both inner sums can be evaluated by known hypergeometric summation theorems. In a similar way, we generate transformations for two-variable hypergeometric series. In the second part we focus on the WZ(Wilf-Zeilberger) method. We use a WZ pair to assign weight to a step and derive the path independence theorem which states that sum of the weights along paths depend only on the endpoints. We derive the change of variable theorem. Then we give some applications of path independence theorem. We also extend the WZ method to Euler's and Pfaff's transformations. We generalized the application of the path independence theorem from hypergeometric functions to symmetric functions in the end.