超几何函数求和式和变换式的推广

Hypergeometric function is a branch of special functions, it is very important in the fields of Combinatorics, Number theory, Li group and the representation of Li Algebra and so on. One of the most important subject of hypergeometric function is the research of summation formulas and transformations which attract many mathematicians, and a lot of famous summation formulas and transformations are found. The object of this program is to generalize and apply the known summations and transformations to get the new ones. There are many methods and techniques for the generalization, here we just present several for examples. Changing the value of the parameters of the hypergeometric function, we can obtain some new summation formulas with the help of the known identities. Usually, we can add pairs of parameters to get some new results. Finding the contiguous relations of hypergeometric function is easy. Comparing the parameters of the hypergeometric functions with different contiguous relations, we can built many new transformations and many of them can be applied on physics, such as the relations of hypergeometric function 3F2 and Appell's F2. In fact, we can get new transformations with the help of summations. Meanwhile, we can obtain new summations by the known transformations. All these new results are the generalization of the summations and transformations of the hypergeometric functions and some of them are useful in the other fields.

超几何函数是一类非常重要的特殊函数，它在组合数学、数论、统计力学、李群及李代数的表示理论、正交函数及偏微分方程的解、域论在物理中的发展等多个方面都有重要应用。超几何函数的求和式和变换式更是具有重要的理论意义和应用价值,从而吸引着众多的科学家,并且大量的求和式和变换式被发现,所以对于超几何函数的求和式和变换式进行了多方位的推广是必要的。通过对超几何函数的参数进行小幅度的整数调整,或者增加参数个数来得到新的超几何函数的求和式,是对原求和式的推广; 从超几何函数的邻近关系式出发, 考虑对应参量的关系, 进而得到新的变换式; 将超几何函数的求和式和变换式相结合, 在计算过程中相互运用, 可以得到更多新的结果来对已有求和式和变换式进行推广和创新.

超几何函数的求和式和变换式是超几何函数的重要研究内容, 随着组合数学和特殊函数的迅猛发展, 更多的求和式和变换式被发现和证明, 这对本学科和相关研究方向的发展都起到了积极的促进作用. 在本项目执行期间, 通过对一些重要的组合计算技巧和超几何函数性质的深入学习和研究, 申请人与合作者对很多著名公式进行了广泛的推广和应用, 如Ramanujan互反公式的推广, Π的级数表达式, q-调和数等相关结果. 从多变量超几何函数的结构出发, 对Horn和Appell的多变量超几何函数的邻近关系式, 变换式和无限求和式等进行分析和研究, 并得到了很多有意义的结果. 相关的研究结果对本学科和相关学科的发展起到了推进的作用.