Bernoulli polynomials, Fourier series and zeta numbers (Q2867702)

This is a very good paper in which the author shows, in an extremely elegant way, how with the aid of the Fourier series for Bernoulli polynomials one can get many interesting relations for the Bernoulli numbers and polynomials, as well as for \(\zeta(2n+1)\) values -- which, after all, is the main goal of this paper. The paper is creative, because the proving techniques, indicated by the author, can be used in much wider and deeper range (for example, by applying some unitary properties of trigonometric Fourier series).NEWLINENEWLINEIn the fourth section, the author introduces the \(p_{2n}(x,a)\) polynomials which play the fundamental role in his series representations of \(\zeta(2n+1)\). No additional information on these ones is presented in the paper. I observe that: NEWLINE\[NEWLINE\begin{aligned} p_{2n}(x,a) :&=2x \frac{B_{2n+1}(x)}{(x^2-a^2)} -\frac{B_{2n+1}(a)}{(x-a)} - \frac{B_{2n+1}(-a)}{(x+a)} \\ &= \frac{B_{2n+1}(x) -B_{2n+1}(a)}{x-a} + \frac{B_{2n+1}(x) - B_{2n+1}(-a)}{x+a}\\ &= \sum_{k=1}^{2n+1} (B_{2n+1}^{(k)}(a)\frac{(x-a)^{k-1}}{k!} +B_{2n+1}^{(k)}(-a)\frac{(x+a)^{k-1}}{k!})\\ &= \sum_{k=1}^{2n+1} \binom{2n+1}{k} (B_{2n+1-k}(a)(x-a)^{k-1} + B_{2n+1-k}(-a)(x+a)^{k-1}). \end{aligned}NEWLINE\]NEWLINE This relation can be used for further discussion on the Scheufens series representation of \(\zeta(2n+1)\).NEWLINENEWLINEReviewer's remark: Many papers of D. Cvijovic and J. Klinowski (as co-authors), and independently of Cvijovic, are of the same character as Scheufens' paper. Furthermore, the Fourier series of Bernoulli polynomials are also discussed in many books concerning analytic number theory, see for example \textit{T. M. Apostol} [Introduction to analytic number theory. New York etc.: Springer-Verlag (1976; Zbl 0335.10001)], \textit{M. Eie} [Topics in number theory. Hackensack, NJ: World Scientific (2009; Zbl 1194.11002)].