An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers

Abstract

Let R=(R,+,·) be a commutative ring of characteristic m>0 (m may be equal to +∞) with unity e and zero 0. Given a positive integer n<m and the so-called n-symmetric set A=a1,a2,…,a2l-1,a2l such that al+i=ne-ai for each i=1,…,l, define the rth power sum Sr(A) as Sr(A)=∑i=12lair, for r=0,1,2,…. We prove that for each positive integer k there holds ∑i=02k-1(-1)i2k-1i22k-1-iniS2k-1-i(A)=0. As an application, we obtain two new Pascal-like identities for the sums of powers of the first n-1 positive integers