基于符号-数值混合计算的线性微分-差分方程的理论及算法研究

A systematic study on the evaluation, integration, summation and further deduction of special functions and infinite sequences is not only of important theoretical interest, but also widely applicable in combinatorics, physics and engineering. The traditional numerical method is efficient and easy to implement, but computational errors are unavoidable. By contrast, symbolic method is accurate and able to provide relatively complete answers for certain classes of functions, however is of high complexity. Under the financial support of our last NSFC fundings (General Program, Young Scholar Fundings) and based on our previous research, we have formulated some creative ideas on how to develop symbolic-numeric hybrid method to study special functions and infinite sequences in more efficient and reliable way. In this project, we will combine the advantages of symbolic and numeric computation to study the theory and algorithms about systems of linear differential-difference equations. On the theoretical aspect, based on Holonomic and Galois methods, we will mainly study three topics: computing Galois groups of linear differential and difference equations, Zeilberger algorithms of multivariate hyperexponential-hypergeometric functions, and evaluation of special functions and infinite sequences. On the algorithmic aspect, we will focus on developing fast and certified algorithms. To achieve this, we will use numerical methods to accelerate the speed and low down the complexity, and will apply symbolic method to validate the numerical results and analyze the error bounds. The above research will greatly improve the capability and efficiency of handling special functions and infinite sequences in practical applications.

特殊函数、无穷序列的赋值、积分、求和运算及推理是一项既有重要理论意义又有很强应用前景的研究。传统的数值方法简便、高效但不可避免计算误差，符号方法完全、准确但计算效率低。通过前期国家自然科学基金青年基金的支持和积累，我们形成了基于符号-数值混合方法研究这类函数的创新性思路。本项目将结合符号计算和数值计算的优点来研究线性微分-差分方程的理论和算法。理论上，我们将以Holonomic、Galois方法为基础，围绕线性微分-差分方程的Galois群计算，多变元超指数-超几何函数的Zeilberger方法以及特殊函数、无穷序列的赋值三个方面展开研究。算法上，重点在于利用数值计算来提高运算速度及降低复杂度，而利用符号分析和验证来检验数值结果并控制误差，发展出快速、可验证的算法，从而进一步提高在组合恒等式的机械化证明以及实际应用中处理特殊函数、无穷序列的效率和能力。

本项目结合符号、数值计算方法来研究线性微分-差分系统的理论和算法。我们根据原定计划，对线性微分-差分系统的 Galois群理论、多变元超指数-超几何函数的Zeilberger方法、特殊函数的赋值分析等问题进行了深入的研究，取得了一些重要进展：首次给出了计算一般线性差分方程Galois群的算法，解决了差分Galois理论中“正问题”这一长期以来的难题；对于D-finite线性微分-差分系统，提出了多变元函数存在邻差算子(telescoper)的刻画，以及一阶含参微分系统的Galois群的计算算法；在特殊函数的赋值研究上，对多伽马函数、误差函数、Bessel函数等特殊函数进行了完整的赋值分析，并结合符号-数值方法发展出了高效可验证算法。