(1)

(2)より、
\begin{align*} B\left(\alpha,\beta\right) & =\left[B\left(z;\alpha,\beta\right)\right]_{z=1}\\ & =\left[\frac{z^{\alpha}}{\alpha}F\left(\alpha,1-\beta;\alpha+1;z\right)\right]_{z=1}\\ & =\frac{1}{\alpha}F\left(\alpha,1-\beta;\alpha+1;1\right) \end{align*}

(2)

\begin{align*} B\left(z;\alpha,\beta\right) & =\int_{0}^{z}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt\\ & =\int_{0}^{z}t^{\alpha-1}\sum_{k=0}^{\infty}C\left(\beta-1,k\right)\left(-t\right)^{k}dt\\ & =\sum_{k=0}^{\infty}C\left(\beta-1,k\right)\left(-1\right)^{k}\int_{0}^{z}t^{\alpha-1+k}dt\\ & =\sum_{k=0}^{\infty}C\left(\beta-1,k\right)\left(-1\right)^{k}\left[\frac{t^{\alpha+k}}{\alpha+k}\right]_{0}^{z}\\ & =\sum_{k=0}^{\infty}C\left(\beta-1,k\right)\left(-1\right)^{k}\frac{z^{a+k}}{\alpha+k}\\ & =z^{\alpha}\sum_{k=0}^{\infty}\frac{P\left(\beta-1,k\right)}{k!}\cdot\frac{\left(-z\right)^{k}}{\alpha+k}\\ & =z^{\alpha}\sum_{k=0}^{\infty}\frac{P\left(\beta-1,k\right)}{k!}\cdot\frac{\left(k+\alpha-1\right)!\left(-z\right)^{k}}{\left(k+\alpha\right)!}\\ & =z^{\alpha}\sum_{k=0}^{\infty}\frac{P\left(\beta-1,k\right)}{k!}\cdot\frac{\left(\alpha-1\right)!Q\left(\alpha,k\right)\left(-z\right)^{k}}{\alpha!Q\left(\alpha+1,k\right)}\\ & =\frac{z^{\alpha}}{\alpha}\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}Q\left(1-\beta,k\right)}{k!}\cdot\frac{Q\left(\alpha,k\right)\left(-z\right)^{k}}{Q\left(\alpha+1,k\right)}\\ & =\frac{z^{\alpha}}{\alpha}\sum_{k=0}^{\infty}\frac{Q\left(\alpha,k\right)Q\left(1-\beta,k\right)}{Q\left(\alpha+1,k\right)}\cdot\frac{z^{k}}{k!}\\ & =\frac{z^{\alpha}}{\alpha}F\left(\alpha,1-\beta;\alpha+1;z\right) \end{align*}