I have discovered (and found an elementary proof of) the following
$$\zeta(2k)=(-1)^{k-1}\dfrac{\pi^{2k}}{2^{4k}-2^{2k}}\left[k\dfrac{2^{2k}}{(2k)!}+{\displaystyle \sum\_{l=1}^{k-1}(-1)^{l}\dfrac{2^{2k-2l}}{(2k-2l)!}\dfrac{2^{4l}-2^{2l}}{\pi^{2l}}\zeta(2l)}\right]$$
($k=1,2,3,\dots$)


My questions are:


1. Is this formula new, or, how can I check if it is?
2. Where should I publish such a result? (if the result or the proof is indeed new)



Is the following recursion formula for $\zeta(2n)$ known?
It's clearest to write the identities in terms of Bernoulli numbers $B\_n$ using
\begin{equation}
\zeta (2 n) = \frac{(-1)^{n - 1} (2 \pi)^{2 n}}{2 (2 n)!} B\_{2 n} .
\end{equation}
Remembering that $B\_1 = \frac{1}{6}$ and that $B\_n$ vanishes for all other odd $n$, the desired identity is
\begin{equation}
\sum\_{k = 0}^{2 n} \binom{2 n}{k} 2^k (2^k - 1) B\_k = 0 .
\end{equation}
In contrast, the identity of Song referred to by Carlo Beenakker is
\begin{equation}
\sum\_{k = 0}^{2 n} \binom{2 n + 1}{k} 2^k B\_k = 0 .
\end{equation}


A useful reference is the book by N. Nielsen, [Traité élémentaire des nombres de Bernoulli](https://gallica.bnf.fr/ark:/12148/bpt6k62119c) . It's necessary to convert notation from Nielsen's book using $B\_n \rightarrow (-1)^{n - 1} B\_{2 n}$, and use the formula for $T\_n$ in equation (9) on page 56. The identity in equation (11) on page 132 is equivalent to
\begin{equation}
\sum\_{k = 0}^{2 n} \binom{2 n}{k} 2^k B\_k + 2 (2^{2 n - 1} - 1)B\_{2 n} = 0 ,
\end{equation}
which is a special case of a more general result attributed to previous authors. The identity in equation (12) on page 138 is equivalent to
\begin{equation}
\sum\_{k = 0}^{2 n} \binom{2 n}{k} 2^{2 k} B\_k + 2 (2^{2 n - 1} - 1)B\_{2 n} = 0 .
\end{equation}
Taking the difference of these gives the desired identity.


The result is also given in equation VII on page 14 of the book by L. Saalschütz, [Vorlesungen über die Bernoullischen Zahlen](https://archive.org/details/vorlesungenberd00saalgoog/page/n28), which references three papers from 1843 by [Stern](https://gdz.sub.uni-goettingen.de/id/PPN243919689_0026?tify=%7B%22pages%22:[94],%22panX%22:0.5,%22panY%22:0.318,%22view%22:%22thumbnails%22,%22zoom%22:0.865%7D), [Schlömlich and Göpel](https://archive.org/details/bub_gb_YxIAAAAAMAAJ/page/n3).


One would expect a number of possible linear identities for the Bernoulli numbers. See, for example, M. W. Coffey, [Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers](https://arxiv.org/abs/1601.01673) and references within.


