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