Consider the following series:
$$S=\sum\_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
$$S=-\frac{5}{16}\Gamma\left(\frac{1}{4}\right)\ { \_3F\_2}\left(1,1,\frac{9}{4};2,2;-1\right).$$
I tried to find an expression of $S$ using elementary functions and ended up with this conjecture:
$$S\stackrel{?}{=}\frac{\Gamma\left(\frac{1}{4}\right)}{16}\Bigg(8\sqrt[4]{8}-16-\ln\frac{24\sqrt{69708+49291\sqrt{2}}+3168\sqrt{2}+4481}{4096}\\\\+\arctan\frac{24\sqrt{49291\sqrt{2}-13260}}{6913}\Bigg).$$My derivation of this formula is quite long and uses a non-rigorous, heuristic approach involving heavy use of techniques like guessing sequence formulas using *Mathematica* command [FindSequenceFunction](http://reference.wolfram.com/mathematica/ref/FindSequenceFunction.html) and OEIS [superseeker](http://oeis.org/demos.html), and recognizing approximate numeric quantities using [RootApproximant](http://reference.wolfram.com/mathematica/ref/RootApproximant.html), [TranscendentalRecognize](https://mathematica.stackexchange.com/a/35/7288), [Inverse Symbolic Calculator](http://isc.carma.newcastle.edu.au) and *Mathematica* command



```
WolframAlpha[ToString[value], IncludePods -> "PossibleClosedForm", TimeConstraint -> ∞]

```

The formula holds with at least $1800$ decimal digits of precision.


Now I am looking for a rigorous way to prove this formula. Can you suggest any ideas how to do that?



An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$
Use formula 16.5.2 from [DLMF](http://dlmf.nist.gov/16.5):
$$\_3F\_2\left(1,1,\frac94;2,2;-1\right)=\int\_0^1{\_2F\_1}\left(1,\frac94;2;-t\right)dt=\frac45\int\_0^1\frac{1-(1+t)^{-5/4}}tdt=\\\\\frac{2}{5}\left(8-4\sqrt[4]{8}+\pi-6\ln2+4 \ln
 (1+\sqrt[4]{2})-4 \arctan\sqrt[4]{2}+2\ln(1+\sqrt2)\right).$$
Some elementary transformations show that this result is equivalent to your conjecture.


