A note on a combinatorial identity (Q1907291)

The following result on a combinatorial identity was obtained by \textit{K. Nagasaka} and \textit{J. S. Shiue} [On a theorem of Koksma on discrepancy, Proc. FIS on algebraic structures and number theory]. Theorem. Let \({\mathbf x} = \{x_n\}^N_{n = 1}\) be a finite sequence of real numbers such that \(0 \leq x_1 \leq x_2 \leq x_3 \leq \cdots \leq x_N < 1\). Then for \(k = 1,2, \dots, N\), \[ \sum^N_{n_1 = 1} \sum^N_{n_2 = 1} \cdots \sum^N_{n_k = 1} \max (x_{n_1}, x_{n_2}, \dots, x_{n_k}) = \sum^N_{n = 1} \psi_k (n) x_n, \] where \[ \psi_k (n) = \sum^k_{i = 1} (-1)^{i + 1} {k \choose i} n^{k - i}. \] We give here a shorter proof for the above theorem. Remark by the editor. K. Nagasaka communicated to the editor that the same proof was also obtained by him and I. Wakabayashi, shortly after the paper had been submitted for publication.