Applications of Zeta function associated with Rational functions of particular forms

• Main Authors: CHAO-NAN HU, 胡照南
• Other Authors: MINKING EIE
• Format: Others
• Language: en_US
• Published: 1999
• Online Access: http://ndltd.ncl.edu.tw/handle/22672926188305591567

碩士 === 國立中正大學 === 應用數學研究所 === 87 === In this paper, we shall produce new Bernoulli identities by considering zeta functions associated with rational functions of the form F(T)=P(T)/(1-Tm1)...(1-Tmr) ,where m1,m2,...mr are positive integers and P(T) is a polynomials in T. We divide the paper into four sections. In section 1, we begin withRiemann zeta function and Hurwitz zeta function as the simplest examples and evaluate these zeta functions at negative integers in terms of Bernoulli numbers andBernoulli polynomials. In section 2, we develop a general method to evaluate zeta functions associated with rational functions. Usually , there are more than one way to express the special values at negative integers. This lead to identities amongBernoulli numbers or Bernoulli polynomials. In section 3 and 4 , we shall provide two main applications of our general theory and produce several newBernoulli identities.