<s>[INST] Evaluate $\sum_{n=1}^{\infty}\frac{\psi^{''}(n)}{2n-1}$, where $\psi^{''}(n)$ is 2nd derivative of digamma function. Does the following sum have a closed form?
$$\sum_{n=1}^{\infty}\frac{\psi^{"}(n)}{2n-1},$$ where $\psi^{"}(n)$ is 2nd derivative of digamma function.
 [/INST] The $n$-th derivate of the digamma function is known as "polygamma function of order $n$-th". Moreover, we have the integral representation
$$\psi^{(m)}(x) = (-1)^{m+1} \int_0^\infty \frac{t^m}{1-e^{-t}} e^{-xt} \, \mathrm{d} t.$$
Using the monotone convergence theorem, we get
$$\label{1} \tag{1}\sum_{n=1}^\infty \frac{\psi''(n)}{2n-1} = - \int_0^\infty \frac{t^2}{1-e^{-t}} \sum_{n=1}^\infty \frac{e^{-nt}}{2n-1} \, \mathrm{d}t. $$
Since
$$\sum_{n=1}^\infty e^{-(n-1/2)t} = \frac{e^{t/2}}{e^t-1},$$
we get by integration
$$ \sum_{n=1}^\infty \frac{e^{-nt}}{2n-1} = \frac{e^{-t/2}}{2} \int_t^\infty \frac{e^{s/2}}{e^s-1} \, \mathrm{d} s.$$
But we have
$$\int_t^\infty \frac{e^{s/2}}{e^s-1} \, \mathrm{d} s = 2 \int_{e^{t/2}}^\infty \frac{1}{x^2-1} \, \mathrm{d} x = 2 \mathrm{arcoth}(e^{t/2}) $$
and therefore
$$\sum_{n=1}^\infty \frac{e^{-nt}}{2n-1} = e^{-t/2} \mathrm{arcoth}(e^{t/2}).$$
Note that we have proven that $$\mathrm{arcoth}(x) = \sum_{n=1}^\infty \frac{x^{-(2n-1)}}{2n-1} \quad \text{for} \quad |x| >1.$$

Thus, we can rewrite \eqref{1} by
  $$\tag{2}\label{2}\sum_{n=1}^\infty \frac{\psi''(n)}{2n-1} = - \int_0^\infty \frac{t^2}{1-e^{-t}} e^{-t/2} \mathrm{arcoth}(e^{t/2}) \, \mathrm{d} t.$$

Changing variables we see that 
\begin{align}
- \int_0^\infty \frac{t^2}{1-e^{-t}} e^{-t/2} \mathrm{arcoth}(e^{t/2}) \, \mathrm{d} t = - 8 \int_1^\infty \frac{\ln(x)^2}{x^2-1} \mathrm{arcoth}(x) \, \mathrm{d} x.
\end{align}
Using $\mathrm{artanh}(1/x)  = \mathrm{coth}(x)$ we can rewrite the integral also by
$$ - 8 \int_1^\infty \frac{\ln(x)^2}{x^2-1} \mathrm{arcoth}(x) \, \mathrm{d} x= 8 \int_0^1 \mathrm{artanh}(y) \frac{\ln(y)^2}{y^2-1} \, \mathrm{d}y.$$

With the help of ComplexYetTrivial's comment we can compute the last integral explicitly.

Using partial integration in the last line we get that
\begin{align}
8 \int_0^1 \mathrm{artanh}(y) \frac{\ln(y)^2}{y^2-1} \, \mathrm{d}y &= \left.-4 \mathrm{artanh}(y)^2 \ln(y)^2 \right|_{y=0}^1 + 8 \int_0\mathrm{artanh}(y)^2 \frac{\ln(y)}{y} \, \mathrm{d} y \\
&= 8 \int_0^1\mathrm{artanh}(y)^2 \frac{\ln(y)}{y} \, \mathrm{d} y.
\end{align}
Since $$\mathrm{artanh}(y) = \frac{1}{2} \left( \ln(y+1)-\ln(1-y) \right) $$
we get that \eqref{2} is equal to
$$2 \int_0^1 \frac{\ln(y+1)^2 \ln(y)}{y} \, \mathrm{d} y - 4 \int_0^1 \frac{\ln(y+1) \ln(1-y) \ln(y)}{y} \, \mathrm{d} y + 2 \int_0^1 \frac{\ln(1-y)^2 \ln(y)}{y} \, \mathrm{d} y $$
The remaining integrals are evaluated here, here and here. In fact, we have
$$2 \int_0^1 \frac{\ln(y+1)^2 \ln(y)}{y} \, \mathrm{d} y = - \zeta(4) = - \frac{\pi^4}{90} $$
and
$$- 4 \int_0^1 \frac{\ln(y+1) \ln(1-y) \ln(y)}{y} \, \mathrm{d} y = \frac{3}{40} \pi^4 - 7 \log(2) \zeta(3) + \frac{\pi^2 \log(2)^2}{3}- \frac{\log(2)^4}{3} - 8 \mathrm{Li}_4(1/2)$$
and also
$$2 \int_0^1 \frac{\ln(1-y)^2 \ln(y)}{y} \, \mathrm{d} y = \frac{\pi^2}{12} - 8 \mathrm{Li}_4(1/2) -7 \log(2) \zeta(3)+ \frac{\pi^2 \log(2)^2}{3} - \frac{\ln(2)^4}{3} $$

Finally, we see that \eqref{2} can be written as
  $$\frac{53}{360} \pi^4 - 14 \log(2) \zeta(3) + \frac{2}{3} [\pi^2 -\log(2)^2] \log(2)^2 - 16 \mathrm{Li}_4(1/2).$$

</s>