Zeta in summations Does anyone know if there is any reference that discuss the following kind of summations?
$$S = \displaystyle\sum_{n=0}^\infty a_n \zeta(n+2)$$
I have read Srivastava's article https://www.sciencedirect.com/science/article/pii/0022247X88900133. But none deals with such series. Especially when:
$a_n = \frac{2}{(n+2)(n+3)}$. Which leads to
$S = \frac{\zeta(2)}{3} + \frac{\zeta(3)}{6} + \frac{\zeta(4)}{10} + \cdots$
Have you seen this one before?
 A: Thanks to a CAS,
$$S_m=\sum_{n=0}^\infty  \frac{2 _,\zeta(n+m)}{(n+m)(n+m+1)}$$ have closed forms
$$S_2=\log (2 \pi )-\gamma$$
$$S_3=\log (2 \pi )-\gamma -\frac{\zeta(2)}{3}$$
$$S_4=\log (2 \pi )-\gamma -\frac{\zeta(2)}{3}-\frac{\zeta (3)}{6}$$
$$S_5=\log (2 \pi )-\gamma -\frac{\zeta(2)}{3}-\frac{\zeta (3)}{6} -\frac{\zeta (4)}{10}$$
$$S_6=\log (2 \pi )-\gamma-\frac{\zeta(2)}{3}-\frac{\zeta (3)}{6} -\frac{\zeta (4)}{10}-\frac{\zeta (5)}{15} $$
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\on}[1]{\operatorname{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
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\begin{align}
S & \equiv \bbox[5px,#ffd]{2\sum_{n = 0}^{\infty}
{\zeta\pars{n + 2} \over \pars{n + 2}\pars{n + 3}}} =
2\sum_{n = 2}^{\infty}{\zeta\pars{n} \over n\pars{n + 1}}
\\[5mm] & =
2\sum_{n = 2}^{\infty}\zeta\pars{n}
\pars{\int_{0}^{1}x^{n - 1}\,\,\dd x}
\pars{\int_{0}^{1}y^{n}\,\dd y}
\\[5mm] & =
2\int_{0}^{1}\int_{0}^{1}
\bracks{\sum_{n = 2}^{\infty}\zeta\pars{n}
\pars{xy}^{n - 1}}y\,\dd x\,\dd y
\\[5mm] & = 
2\int_{0}^{1}\int_{0}^{1}
\bracks{-\gamma - \Psi\pars{1 - xy}}y\,\dd x\,\dd y
\end{align}
$\ds{\gamma}$ is the Euler-Mascheroni Constant and $\ds{\Psi}$ is the Digamma Function.
The relation between $\ds{\zeta\ \mbox{and}\ \Psi}$ can be seen in
$\ds{\color{black}{\bf 6.3.14}}$ of
$\mbox{A & S Table}$.
Then,
\begin{align}
S & \equiv \bbox[5px,#ffd]{2\sum_{n = 0}^{\infty}
{\zeta\pars{n + 2} \over \pars{n + 2}\pars{n + 3}}}
\\[5mm] & =
2\int_{0}^{1}
\bracks{-\gamma -
{\ln\pars{\Gamma\pars{1 - y}} \over - y}}y\,\dd y
\\[5mm] & =
2\bracks{-\,{1 \over 2}\,\gamma +
\int_{0}^{1}\ln\pars{\Gamma\pars{1 - y}}\dd y}
\\[5mm] & =
\bbx{\ln\pars{2\pi} - \gamma} \approx 1.2607 \\ &
\end{align}

I used the well known result
$\ds{\int_{0}^{1}\ln\pars{\Gamma\pars{1 - y}}\,\dd y
= {\ln\pars{2\pi} \over 2}}$
A: Since
$$
\zeta (s) = {1 \over {\Gamma (s)}}\int_0^\infty  {{{x^{\,s - 1} } \over {e^{\,x}  - 1}}dx} 
$$
then
$$
\eqalign{
  & \sum\limits_{n = 0}^\infty  {a_{\,n} \zeta (n + 2)}
  = \sum\limits_{n = 0}^\infty  {a_{\,n} {1 \over {\left( {n + 1} \right)!}}\int_0^\infty  {{{x^{\,n + 1} }
 \over {e^{\,x}  - 1}}dx} }  =   \cr 
  &  = \int_0^\infty  {\left( {\sum\limits_{n = 0}^\infty  {a_{\,n} } {{x^{\,n + 1} }
 \over {\left( {n + 1} \right)!}}} \right){1 \over {e^{\,x}  - 1}}dx}  =   \cr 
  &  = \int_0^\infty  {{{I(x)} \over {e^{\,x}  - 1}}dx}  \cr} 
$$
if we admit that the integral and sum can be exchanged.
Here
$$
I(x) = \sum\limits_{n = 0}^\infty  {a_{\,n} } {{x^{\,n + 1} } \over {\left( {n + 1} \right)!}}
 = \int_0^x {\left( {\sum\limits_{n = 0}^\infty  {a_{\,n} } {{t^{\,n} } \over {n!}}} \right)dt} 
$$
is the integral of the e.g.f of the $\{a_n\}$ sequence.
