Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}\_2F\_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) \alpha B(2,\alpha +1)}
$$
and
$$
{}\_2F\_1(2\alpha+1,\alpha+1;\alpha+3;-1)
=\frac{1}{2^{2\alpha}}\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha)\alpha B(2,\alpha+1)},
$$
where the Beta function is denoted and defined by
$$
B(z,w)=\int\_0^1t^{z-1}(1-t)^{w-1}\textrm{d}t
=\int\_0^\infty\frac{t^{z-1}}{(1+t)^{z+w}}\textrm{d}t
$$
for $\Re(z),\Re(w)>0$ and the Gauss hypergeometric function ${}\_2F\_1$ is defined by
$$
{}\_pF\_q(\alpha\_1,\dotsc,\alpha\_p;\beta\_1,\dotsc,\beta\_q;z)
=\sum\_{n=0}^\infty\frac{(\alpha\_1)\_n\cdots(\alpha\_p)\_n} {(\beta\_1)\_n\cdots(\beta\_q)\_n}\frac{z^n}{n!}
$$
for $\alpha\_i\in\mathbb{C}$, $\beta\_i\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$, $p,q\in\mathbb{N}$, and $z\in\mathbb{C}$, in terms of the rising factorial
$$
(z)\_n=\prod\_{\ell=0}^{n-1}(z+\ell)
=
\begin{cases}
z(z+1)\dotsm(z+n-1), & n\in\mathbb{N};\\
1, & n=0.
\end{cases}
$$



Ask for references or proofs of two explicit formulas for special Gauss hypergeometric functions
These formulas are special cases of what I like to call
extendable evaluations: set for simplicity $F={}\_2F\_1$. If one
has an explicit formula for $F(a,b;c;z)$ and (for example) also for
$F(a+1,b;c;z)$, then using the contiguity relations, it is immediate
to find explicit formulas for $F(a+k,b+l;c+m;z)$ for any
$(k,l,m)\in {\mathbb Z}^3$. For $z=-1$ there is the classical Kummer evaluation
$$F(a,b;a-b+1;-1)=\dfrac{2^{-a}\Gamma(1/2)\Gamma(a-b+1)}{\Gamma(a/2+1/2)\Gamma\
(a/2-b+1)}\;,$$
and the less classical (sorry I do not remember a good reference)
\begin{align\*}&F(a+1,b;a-b+1;-1)=2^{-a-1}\Gamma(1/2)\Gamma(a-b+1)\ \cdot\\
 &\phantom{=}\cdot\left(\dfrac{1}{\Gamma(a/2+1/2)\Gamma(a/2-b+1)}+\dfrac{1}{\Gamma(a/2+1)\Gamma(a/2-b+1/2)}\right)\;,\end{align\*}
so this is extendable and in particular immediately gives your formula for
$z=-1$. Similarly, there are two extendable two-parameter families
for $z=1/2$ and two for $z=2$.


