Is there any simpler form of this function Assume that $n$ is a positive integer. Is there any simple form of this hypergeometric value $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$
 A: write $F(n)$ for your formula.  Then
$$
F \left( 0 \right) =\frac{1}{4}\,\pi \\
F \left( 2 \right) =-\frac{3}{2}+\frac{3}{4}\,\pi \\
F \left( 4 \right) =-5+{\frac {15}{8}}\,\pi \\
F \left( 6 \right) =-{\frac {77}{6}}+{\frac {35}{8}}\,\pi \\
F \left( 8 \right) =-30+{\frac {315}{32}}\,\pi \\
F \left( 10 \right) =-{\frac {671}{10}}+{\frac {693}{32}}\,\pi 
$$
and
$$
F \left( 1 \right) =-2+2\,\sqrt {2}\\
F \left( 3 \right) =-{\frac {20}{3}}+\frac{16}{3}\,\sqrt {2}\\
F \left( 5 \right) =-{\frac {86}{5}}+{\frac {64}{5}}\,\sqrt {2}\\
F \left( 7 \right) =-{\frac {1416}{35}}+{\frac {1024}{35}}\,\sqrt {2}\\
F \left( 9 \right) =-{\frac {5734}{63}}+{\frac {4096}{63}}\,\sqrt {2}\\
F \left( 11 \right) =-{\frac {46124}{231}}+{\frac {32768}{231}}\,
\sqrt {2}
$$
Now, can you guess the patterns?
A: First apply the Pfaff transformation mentioned above:
$$
F(2n)=1/2\  _2F_1(1,n+1;\frac{(n+1)+n}{2};1/2).
$$
Then use a formula from section 7.3.8 in Prudnikov,Bychkov,Marichev Integral and series, vol.3 to represent $F(2n)$ in terms of $\Gamma$ functions and a finite sum $\sum_1^n$.
(In my Russian edition of this vol.3 it is formula 15, section 7.3.8, page 415).
After Pfaff transformation it is also possible to derive rather simple integral form using PBM-3 section 7.3.1 in terms of incomplete Beta functions. 
