Speaker: Christoph Koutschan，奥地利科学院RICAM研究所

Title: Identity Found by Proving Identities

Language: Chinese

Time & Venue: 2025.08.14 15:00-16:00 思源楼 525

Abstract: At the third edition of FPSAC, which took place in 1991 in Bordeaux, Doron Zeilberger gave an invited talk with the title "Identities in Search of Identity". Shortly before he had published his seminal paper on the holonomic systems approach to special function identities and, together with Herbert Wilf, had developed a framework for proving hypergeometric summation identities, nowadays known as WZ theory. During the following 35 years, the theory has been considerably extended and refined, and evolved into its own research area within symbolic computation. The corresponding implementations in computer algebra systems can help to solve various summation and integration problems, and likewise they support many operations on formal power series. Therefore it is not surprising that this algorithmic framework has found numerous applications, for instance in combinatorics. In our lecture, we aim at recapitulating the evolution of this research area, highlighting its main achievements, and discussing some recent trends. A special emphasis is put on the treatment of determinants and Pfaffians that became also 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.