An identity in commutative rings with unity with applications to various sums of powers (Q2398750)

Summary: Let \(R = (R, +, \cdot)\) be a commutative ring of characteristic \(m > 0\) (\(m\) may be equal to \(+ \infty\)) with unity \(e\) and zero 0. Given a positive integer \(n < m\) and the so-called \(n\)-symmetric set \(A = \left\{a_1, a_2, \ldots, a_{2l-1}, a_{2l} \right\}\) such that \(a_{l+i} = ne-a_i\) for each \(i = 1, \ldots, l\), define the \(r\)th power sum \(S_r(A)\) as \(S_r(A) = \sum_{i = 1}^{2 l} a_i^r\), for \(r = 0,1, 2, \ldots \). We prove that for each positive integer \(k\) there holds \[ \sum_{i = 0}^{2k-1}(-1)^i \binom{2k-1}{i} 2^{2k-1-i} n^i S_{2k-1-i}(A) = 0. \] As an application, we obtain two new Pascal-like identities for the sums of powers of the first \(n-1\) positive integers.