A few years ago I was working on a theory for superconductivity inspired by a classic paper by Lev Gor'kov from 1959. At some point I needed to benchmark the theory in a known case, so I considered the theory under the same conditions as Gor'kov. When I subsequently evaluated an integral I encountered, by invoking some known series expansions, I decided to equate my result (a sum) with the result Gor'kov obtained (bluntly hoping that the results would be equal). In doing so, a series expansion for $\zeta(3)$ popped out:
$$\zeta(3) = -\frac{8\pi^2}{3} \sum_{n=0}^{\infty} \frac{\zeta(2n) }{(2n+1)(2n+2)(2n+3)}. $$
I have not seen this in the published literature, so I was wondering if anyone here has seen it before or has any suggestions on how to formally prove it. I know that many similar (and more rapidly converging) series are known, e.g., as discovered by Chen and Srivastava.
Furthermore, I played around with some generalizations of the above formula and reached the following:
$$ \zeta(2n+1) = \pi^{2n} \sum_{m = 1}^n b_{m,n} (-1)^m \frac{2^{2m+1}}{2m+1} \sum_{k=0}^{\infty} \frac{\zeta(2k)}{\prod_{j=1}^{2m+1}(2k+j)}, $$
wherein the weights $b_{m,n}$ appear to be rational and positive, $b_{m,n} \in \mathbb{Q}^+ $ and where $ b_{n,n} = 1$. For instance, $b_{1,1} = 1$, $\lbrace{b_{m,2}\rbrace}_m = \lbrace \frac{4}{15}, 1 \rbrace$, $\lbrace{b_{m,3}\rbrace}_m = \lbrace \frac{4}{63}, \frac{8}{21}, 1 \rbrace$, $\lbrace{b_{m,4}\rbrace}_m = \lbrace \frac{8}{525}, \frac{104}{945}, \frac{4}{9}, 1 \rbrace$, $\lbrace{b_{m,5}\rbrace}_m = \lbrace \frac{604}{155925}, \frac{944}{31185}, \frac{212}{1485}, \frac{16}{33}, 1 \rbrace$. I would be grateful to receive any references on it if has been published and any advice on how to formally prove it. The above formula is reminiscent of the more rapidly converging series found by D. Cvijović and J. Klinowski.
