The structural elucidation of Eisenstein's formula  

作　　者：HASHIMOTO Masahiro KANEMITSU Shigeru 

机构地区：[1]Department of Mathematics,Wakayama College of Technology,Noda-machi Gobo-shi 649-15,Japan [2]Graduate School of Advanced Technology,Kinki University Iizuka,Fukuoka 820-8555,Japan

出　　处：《Science China Mathematics》2010年第9期2341-2350,共10页中国科学：数学（英文版）

基　　金：supported by Natural Science Foundation of Shaanxi Province (Grant Nos.SJ08A22,2010JM1009)

摘　　要：In this paper,we shall give a complete structural description of generalizations of the classical Eisenstein formula that expresses the first periodic Bernoulli polynomial as a finite combination of cotangent values,as a relation between two bases of the vector space of periodic Dirichlet series.We shall also determine the limiting behavior of them,giving rise to Gauss' famous closed formula for the values of the digamma function at rational points on the one hand and elucidation of Eisenstein-Wang's formulas in the context of Kubert functions on the other.W shall reveal that most of the relevant previous results are the combinations of the generalized Eisenstein formula and the functional equation.In this paper,we shall give a complete structural description of generalizations of the classical Eisenstein formula that expresses the first periodic Bernoulli polynomial as a finite combination of cotangent values,as a relation between two bases of the vector space of periodic Dirichlet series.We shall also determine the limiting behavior of them,giving rise to Gauss’ famous closed formula for the values of the digamma function at rational points on the one hand and elucidation of Eisenstein-Wang’s formulas in the context of Kubert functions on the other.W shall reveal that most of the relevant previous results are the combinations of the generalized Eisenstein formula and the functional equation.

关 键 词：Eisenstein FORMULA HURWITZ ZETA-FUNCTION POLYLOGARITHM FUNCTION GAUSS FORMULA 

分 类 号：O174.14[理学—数学]