MacMahon’s Partition Analysis IV: Hypergeometric Multisums

In his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis as a computational method for solving problems in connection with lin-ear homogeneous diophantine inequalities and equations, respectively. The object of this paper is to introduce an entirely new application domain for MacMahon’s operator technique. Namely, we show that Partition Analysis can be also used for proving hypergeometric multisum identities. Our examples range from combina-torial sums involving binomial coefficients, harmonic and derangement numbers to multisums which arise in physics and which are related to the Knuth-Bender theorem