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 $$ B(z,w)=\int_0^1t^{z1}(1t)^{w1}\textrm{d}t =\int_0^\infty\frac{t^{z1}}{(1+t)^{z+w}}\textrm{d}t $$ 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 $$ (z)_n=\prod_{\ell=0}^{n1}(z+\ell) = \begin{cases} z(z+1)\dotsm(z+n1), & n\in\mathbb{N};\\ 1, & n=0. \end{cases} $$

$\begingroup$ Please read the paper "YueWu Li and Feng Qi, A new closedform formula of the Gauss hypergeometric function at specific arguments, Axioms 13 (2024), no. 5, Article 317, 24 pages; available online at doi.org/10.3390/axioms13050317." $\endgroup$– qifeng618Commented May 10 at 15:32
1 Answer
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 $$F(a,b;ab+1;1)=\dfrac{2^{a}\Gamma(1/2)\Gamma(ab+1)}{\Gamma(a/2+1/2)\Gamma\ (a/2b+1)}\;,$$ and the less classical (sorry I do not remember a good reference) \begin{align*}&F(a+1,b;ab+1;1)=2^{a1}\Gamma(1/2)\Gamma(ab+1)\ \cdot\\ &\phantom{=}\cdot\left(\dfrac{1}{\Gamma(a/2+1/2)\Gamma(a/2b+1)}+\dfrac{1}{\Gamma(a/2+1)\Gamma(a/2b+1/2)}\right)\;,\end{align*} so this is extendable and in particular immediately gives your formula for $z=1$. Similarly, there are two extendable twoparameter families for $z=1/2$ and two for $z=2$.

$\begingroup$ Please read the paper "YueWu Li and Feng Qi, A new closedform formula of the Gauss hypergeometric function at specific arguments, Axioms 13 (2024), no. 5, Article 317, 24 pages; available online at doi.org/10.3390/axioms13050317." $\endgroup$ Commented May 10 at 15:30