Finding sum of powers on arithmetic progressions with application of Cauchy's equation (Q1412405)

Let \(S_k(n;a,h)=a^k+(a+h)^k+\dots+(a+(n-1)h)^k\) for positive integers \(k,n\) and reals \(a,h\). \textit{J.~Aczél} [Aequationes Math. 21, 39--43 (1980; Zbl 0451.39006)] obtained the general sum as \[ S_k(n;a,h)=\frac{1}{k+1}\sum_{j=0}^k \binom{k+1}{j+1}n^{j+1}c_{k-j}h^j,\tag{1} \] where \(c_l\) satisfy \( \sum_{j=1}^l\binom{l}{j}c_{l-j}h^{j-1}=la^{l-1}\), \(l=1,2,\dots\). In this paper the authors give an alternative proof of (1) based on induction. For \(k=4\) they also present a proof of (1) with the help of the additive Cauchy equation. Applying (1) the authors determine some other sums as: \(-1^2-2^2+3^2-\dots -(3n-2)^2-(3n-1)^2+(3n)^2=-3n^3 +(9/2)n^2+(5/2)n\); \(1.2.3+2.3.4+\dots+n.(n+1).(n+2)=(1/4)n(n+1)(n+2)(n+3)\); \(1.3.5+2.4.6+\dots+(n-2).n.(n+2)=(1/4)(n-2)(n-1)(n+2)(n+3)\). Finally, following Aczél [loc. cit.], they consider (1) on a divisible commutative ring of a finite characteristic \(m\) and express \(S_k(n;a,h)\) as polynomials with the powers \(a^{k-i}h^i\), \(i=1,2,\dots,k\), adding \(a^k\) if \(n\) is not divisible by \(m\). For \(m=2\) the authors prove that all coefficients of such powers of \(a\) and \(h\) in \(S_k(n;a,h)\) are nonzero if and only if \(k=1\) or \(k=k_i\), where \(k_1=3\) and \(k_{i+1}=2k_i+1\), \(i=1,2,\dots\).