Hypergeometric series

Abstract

Hypergeometric series form a particularly powerful class of series, as they appear in a great variety of different scientific contexts while at the same time allowing a rather simple definition. Popularized by the work of Gauss, hypergeometric series have been intensively studied since the 19th century, and they are still subject of ongoing research. Nowadays, they are also well understood from an algorithmic point of view, and in this chapter, we will see some of the most important algorithms for dealing with them. 5.1 The Binomial Theorem The binomial theorem states that for n E N and any a, b E lK we have (a+bt = ± (n)akb n-k. k=O k If this is to be a theorem, there should be a proof. In order to give a rigorous proof, we need to go back to the definition of the symbol CD. Where the binomial coefficients appeared in earlier chapters of this book, we have silently considered them as known and did not give a formal definition. Now it is time to close this gap. There are several possibilities. We could simply take the binomial theorem itself as the definition. We could as well declare that G) be defined recursively via the Pascal triangle recurrence (n,k > 0) together with the boundary conditions (~) = I (n ~ 0) and (~) = 0 (n > 0). We could instead also take a combinatorial approach and decide that G) should be the number of subsets of {I, 2, ... , n} with exactly k elements (hence the pronunciation M. Kauers, P. Paule, The Concrete Tetrahedron