Contiguous Relations and Creative Telescoping
Contiguous relations are a fundamental concept within the theory of hypergeometric series and orthogonal polynomials. Their study goes back to Gauss who gave a list of 15 `fundamental' relations for the 2F1 case. Applications range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series. Creative telescoping is the underlying principle of Zeilberger's extension of Gosper's algorithm. The resulting algorithm for definite summation of terminating hypergeometric series constitutes a major break-through in symbolic summation.
Besides surveying these concepts, the main theme of the talk is to establish a new connection between them. Namely, all classical contiguous relations between terminating or non-terminating hypergeometric series can be computed by creative telescoping. This, for instance, allows computer proofs of summation formulas involving also non-terminating hypergeometric sums, or the automatic discovery of new recurrences for given combinatorial sums. Various illustrative examples derived by corresponding computer algebra programs are given.