Hypergeometric Functions

About one-sixth of all the identities on the site contain hypergeometric functions. Many of them occur in the general form for arbitrary positive integers and . This means that each of these identities encodes a whole family of identities for , , , , , , . In the building phase of the hash tables, the most common instances are explicitly generated and then analyzed.

It often happens (for instance as a result of Integrate) that you want to reduce a hypergeometric function to simpler functions like elementary functions, Bessel functions, error functions, and so on. The site contains thousands of such special cases. Because the first and second arguments can be interchanged and the elements in each argument group can be reordered, finding these specializations can be time-consuming. We wrote a special purpose search code for these cases. Given a pattern of a hypergeometric function, say , the four possible pairs of realizations , , ... are constructed and matched against all structurally matching hypergeometric functions in the collection. For each such realization and each member of the collection, Reduce is called to find out if a match is possible. No explicit realizations are needed, but impossible matches must be ruled out. While this is a relatively time-consuming procedure (of the order of seconds for a typical hypergeometric function), it finds all hypergeometric identities that match a certain pattern. See Search 7 for an example.