Seminar Details

At the 3rd Formal Power Series and Algebraic Combinatorics conference, that tool place in Bordeaux in 1991, Doron Zeilberger gave an invited talk with the title "Identities in Search of Identity". At the same time, his seminal paper on the holonomic systems approach to special function identities was published and, together with Herbert Wilf, he developed the WZ theory for proving hypergeometric summation identities. During the following 35 years, this theory has been considerably extended and refined, and evolved into its own research area within symbolic computation. We recapitulate its evolution, highlight its main achievements, and discuss some recent trends. We then turn our attention to applications in combinatorics, with a special emphasis on the treatment of determinants and Pfaffians, which became amenable to symbolic methods via the holonomic ansatz: the sought identity may be transformed into a set of summation identities, which themselves can be proven algorithmically. This procedure is elucidated with prominent examples, such as the q-enumeration of totally-symmetric plane partitions, the counting of configurations in the twenty-vertex model, and the evaluation of binomial determinants emerging from rhombus tilings.