关于一致函数的一个互反律

摘要

设F是有两个复变元的到一个加法Abel群中的一致函数,即F满足孙智伟在1989年引入的下述函数方程∑n-1 r=0 F(x+r,ny) = F(x,y), n = 1,2,3,….假定∈Dom(F)时也有∈Dom(F).我们建立了下述互反律:∑m-1 r=0 F(x+nr/m,my) = ∑n-1 r=0 F(x+mr,ny) ( ∈Dom(F),m,n=1,2,3,…).文中还给出它的几个应用.%Let F be a uniform function of two complex variables to an additive abelian group,i.e. , F satisfies the functional equation introduced by Z. W. Sun in 1989. Suppose that ∈Dom(F) for all ∈ Dom(F) . In this paper we establish the following reciprocity law:m-1 m-1 ∑F(x + nr/m,my) = ∑F(x+mr,ny)r=0 r=0 for any ∈ Dom(F) and m,n=1,2,3,…. Several applications are also given.