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# What are Wilf-Zeilberger pairs?

Alexander Kemarski will give a talk about Wilf-Zeilberger pairs.

Given an infinite hypergeometric series it is possible to decide algorithmically whether there is a nice result for the sum of the series and if not, prove that there is no simple formula for the sum of the series. In the talk I will describe basic definitions and some theorems in this topic, the proofs can be found in http://www.springer.com/la/book/9781447164630.

For example, the sum of binomial coefficients C(n,k) over k has a nice expression 2**n, the sum of binomial coefficients C(n,k)**2 has a nice expression C(2n,n), but it can be proven that there is no closed expression for the sum of binomial coefficients C(n,k)**3.

Remark: the class of hypergeometric functions is very big and includes all familiar functions from the calculus courses. For example, K-finite matrix coefficients of representations of Lie groups are hypergeometric functions https://rd.springer.com/article/10.1007/s10688-007-0027-6.

This lecture is part of the "What is..?" seminar.