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The question: Is the following formula known?

$$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+2}-1)B_{2v-2k+2} B_{2m-2v}(\frac{1}{2})}{(2^{2v+1}-1)(2v-2k+2)!(2m-2v)!}$$ with $m\in\mathbb{N}$, the Bernoulli numbers $B_k$ and the coefficient $b_{2k}:=\frac{1}{2k}-\ln 2-\sum\limits_{v=1}^\infty \frac{\zeta(2v)}{2^{2v} (k+v)}$.

Some informations:

The basis for the derivation of the formula above is the recursion $a_m+ \sum\limits_{v=1}^m \binom{2m}{2v}(1-\frac{1}{2^{2v}})a_v =b_{2m}$ with $a_m:=(-1)^m \frac{(2m)!\zeta(2m+1)}{\pi^{2m}}$ and the definition of $b_{2m}$ above.

The derivation of this recursion is too much for a forum, hence I would like to point only to an already existing recursion developed by Kurokawa (" Multiple sine functions and Selberg zeta functions" ): http://projecteuclid.org/euclid.pja/1195512182 , page 62 (4) and page 63 above

See https://www.fernuni-hagen.de/analysis/download/bachelorarbeit_aschauer.pdf , page 26, No. (6), proof at page 29 .

The definition of function $Q_m(x)$ is on page 13, (1), a variation of the Multiple Gamma-function.

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  • $\begingroup$ How do you evaluate $b_{2k}$? $\endgroup$
    – მამუკა ჯიბლაძე
    Apr 21, 2016 at 12:09
  • $\begingroup$ That's what I meant with "The derivation of this recursion is too much for a Forum, ...". It was a result of my Bachelor-Thesis. But the script of Kurokawa shows more or less the same by using page 62 (4) to get the recursion on page 63. $\endgroup$
    – user90369
    Apr 21, 2016 at 12:28
  • $\begingroup$ Well Kurokawa does not say anything about convergence, at least I could not find it. He expresses his $F_{2k+1}(\frac12)$ through values of Barnes multiple Gamma functions but does not say much about ways to calculate the latter. Are your $b_{2m}$ expressible through some known special functions too? $\endgroup$
    – მამუკა ჯიბლაძე
    Apr 21, 2016 at 13:01
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    $\begingroup$ Series expansion of Kurokawas Multiple sine function for $r\ge 2$: $\ln F_r(z)=\frac{z^{r-1}}{r-1}+(-1)^r 2 \sum\limits_{v=1}^\infty \frac{(-z)^{r-1+2v}\zeta(2v)}{r-1+2v}$ $\endgroup$
    – user90369
    Apr 22, 2016 at 10:06
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    $\begingroup$ @მამუკა ჯიბლაძე : I have added the link of the origin paper. $\endgroup$
    – user90369
    Aug 19, 2016 at 7:21

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