Hypergeometric Summation Algorithms for High-order Finite Elements

High-order finite elements are usually defined by means of certain orthogonal polynomials. The performance of iterative solution methods depends on the condition number of the system matrix, which itself depends on the chosen basis functions. The goal is now to design basis functions minimizing the condition number, and which can be computed efficiently. In this paper, we demonstrate the application of recently developed computer algebra algorithms for hypergeometric summation to derive cheap recurrence relations allowing a simple implementation for fast basis function evaluation.