多变元Zeilberger算法及其应用

Zeilberger算法是组合恒等式机器证明理论的核心算法，也是解决组合数学、数论和数学物理等领域涉及复杂求和问题的重要计算工具。该算法是针对超几何项的单求和问题设计，为了有效处理数学中广泛存在的各类多重求和问题，迫切需要发展多变元的相关理论与算法。围绕该课题，本项目的主要研究内容如下：.（1）通过Nicole定理的引入与推广，将多变元有理函数的Zeilberger算法用于涉及非完整函数与调和数的多重求和恒等式的证明。.（2）综合运用近期发展的离散留数和轨道分解等工具，解决多变元超几何项的可求和性问题并设计可以构造极小邻差算子的多变元超几何项Zeilberger算法。.（3）基于多项式约化后函数的可求和性，将多变元Zeilberger算法用于涉及多重求和同余式的自动证明。.本项目的顺利实施将进一步丰富组合恒等式机器证明的理论、算法及应用。

Zeilberger's algorithm is a key algorithm in the theory of mechanical proofs of combinatorial identities, which has become a powerful algorithmic tool in dealing with complicated summations from combinatorics, number theory and mathematical physics. Nevertheless, Zeilberger's original algorithm was designed for simple sums of hypergeometric terms. In order to deal with multiple summations automatically, it is natural to extend the algorithm to the multivariate case. For this purpose, this project mainly focuses on the following three topics:.(1) Application of the multivariate rational Zeilberger's algorithm to the proofs of identities involving non-holonomic functions and harmonic numbers with the assistance of Nicole's theorem..(2) Extending Zeilberger's algorithm to the multivariate hypergeometric case, namely, solving the summability problem and designing an algorithm to construct minimal telescopers, with the newly developed discrete residue and orbit decomposition theory..(3) Mechanical proofs of congruence identities utilizing the multivariate Zeilberger's algorithm and polynomial reduction..This project will further enrich the theory, algorithms and applications of mechanical proofs of combinatorial identities.