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

Abstract EN

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

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.