多参数超几何级数恒等式与一些重要数学常数的无穷级数表示

Hypergeometric series identities play an important role in combinatorics, number theory, theoretical physics and computer algebra. It is particularly significant to find new summation and tranformation formulae. In practical applications, we use only the approximate values of irrational numbers which can be defined as the infinite non-repeating decimals. Therefore researching series expansions of irrational numbers and their reciprocal are significant. The hypergeometric method is an effective method to derive the expansions of irrational numbers. The hypergeometric series identities with more parameters are more flexible and can reduce faster convergent series expansions. Based on the applicant’s research work on well-poised and quadratic hypergeometric series identities with more parameters, the project will further investigate systematically the hypergeometric series identities with more parameters and their application in faster convergent series expansions of ratio of the circumference, Euler-Mascheroni constant and other irrational numbers about the two constants.

超几何级数理论中许多恒等式在组合数学、数论、理论物理及计算机代数等领域都有重要应用．寻找和证明新的求和公式和变换关系是超几何级数领域的主要研究课题．无理数即无限不循环小数，在实际应用中，我们往往只使用其近似值，从这一角度考虑，探讨无理数的无穷级数表示具有好的实际意义．超几何方法是研究无理数级数展开的有效方法，含自由参数越多的超几何级数恒等式应用起来越灵活，也越可能产生收敛速度更快的级数展开，但现有多参数的超几何级数恒等式相对较少．基于申请人已完成的对列平衡和二次超几何级数的多参数推广工作，本项目旨在进一步系统研究含有较多自由参数的超几何级数恒等式，并应用其建立圆周率、Euler-Mascheroni常数等重要数学常数以及与之相关的无理数的快速收敛的级数展开式.

超几何级数恒等式在组合数学、数论、理论物理以及计算机代数等领域都有重要应用。无理数的无穷级数展开对于其近似值的研究有很好的意义，超几何级数方法是建立无理数无穷级数展开式的有效方法。含自由参数越多的超几何级数恒等式应用起来越灵活，也越可能产生收敛速度更快的级数展开．在本项目中，我们利用Abel分部求和引理建立了新的含有多个自由参数的三次超几何级数求和公式和变换公式，并通过这些结果来推导与圆周率相关的无理数的级数展开式.