Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}_2F_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=\frac{2[\alpha  B(1/2,\alpha)-1-\alpha]}{(1-\alpha) \alpha  B(2,\alpha +1)}
$$
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
$$
(z)_n=\prod_{\ell=0}^{n-1}(z+\ell)
=
\begin{cases}
z(z+1)\dotsm(z+n-1), & n\in\mathbb{N};\\
1, & n=0.
\end{cases}
$$