设{x_i}为任一复数序列。从两个初等代数恒等式的证明出发,研究了一类求和运算的封闭形式.基本定理可叙述如下;记{θ_j}p 为 p 次单位根,则有■通过对序列{x_j)和变量 t ,τ的特殊选择,上述定理给出一系列关于二项式系数及 Gauss 二项...设{x_i}为任一复数序列。从两个初等代数恒等式的证明出发,研究了一类求和运算的封闭形式.基本定理可叙述如下;记{θ_j}p 为 p 次单位根,则有■通过对序列{x_j)和变量 t ,τ的特殊选择,上述定理给出一系列关于二项式系数及 Gauss 二项式系数的求和公式。其中包括徐利治、欧阳植(1984~1985)的新近工作作为特款。此外,定理的极限形式还可给出 Euler 关于自然数例数偶次幂和公式的一种新的推导。展开更多
The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-ser...The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).展开更多
文摘设{x_i}为任一复数序列。从两个初等代数恒等式的证明出发,研究了一类求和运算的封闭形式.基本定理可叙述如下;记{θ_j}p 为 p 次单位根,则有■通过对序列{x_j)和变量 t ,τ的特殊选择,上述定理给出一系列关于二项式系数及 Gauss 二项式系数的求和公式。其中包括徐利治、欧阳植(1984~1985)的新近工作作为特款。此外,定理的极限形式还可给出 Euler 关于自然数例数偶次幂和公式的一种新的推导。
文摘The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).