Speaker：Qinhu Hou（Tianjin University）

Time：2022-09-21, 10:30

Location：Tencent Meeting ID：947-841-4036 （Password：260172）

Abstract:

In this talk, we will traduce the basic ideas and classical algorithms for the automatic proofs of combinatorial identities, including Sister Celine’s method, Gosper’s algorithm, Zeilberger’s algorithm, etc. 

It was Sister Celine who firstly gave the automatic method of computing the recurrence relation of summations, which is the mile stone in this field. Gosper’s algorithm completely solves the problem of indefinite summation of hypergeometric terms and is the key algorithm. Zeilberger’s algorithm provides a fast method for computing the recurrence relation of definite summation. Moreover, one can construct new identities by WZ method.

We will also introduce some recent development and unsolved problems in this field. One generalization of hypergeometric terms is the P-recursive sequences, which can be treated similar to Zeilberger’s algorithm. By developing Karr’s theory, we can also deal with summations involving harmonic numbers. For some identities in q-series, we can also verify them by the theory of modular forms.

At last, we will give a brief introduction on the packages and examples. There are more than one implements of Gosper’s algorithm and Zeilberger’s algorithm, including packages in Maple and Mathematica. We will illustrate the use of these packages by proving some congruences.