(1)

\begin{align*} xB(x,y+1) & =x\frac{\Gamma(x)\Gamma(y+1)}{\Gamma(x+y+1)}\\ & =y\frac{\Gamma(x+1)\Gamma(y)}{\Gamma(x+y+1)}\\ & =yB(x+1,y) \end{align*}

(2)

\begin{align*} B(x,y) & =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\\ & =\frac{y-1}{x}\frac{\Gamma(x+1)\Gamma(y-1)}{\Gamma(x+y)}\\ & =\frac{y-1}{x}B(x+1,y-1) \end{align*}

(2)-2

\begin{align*} B(x,y) & =\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt\\ & =\left[\frac{1}{x}t^{x}(1-t)^{y-1}\right]_{t=0}^{t=1}+\frac{y-1}{x}\int_{0}^{1}t^{x}(1-t)^{y-2}dt\\ & =\frac{y-1}{x}B(x+1,y-1) \end{align*}

(3)

\begin{align*} B(x,y) & =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\\ & =\frac{x+y}{x}\frac{\Gamma(x+1)\Gamma(y)}{\Gamma(x+y+1)}\\ & =\frac{x+y}{x}B(x+1,y) \end{align*}

(4)

\begin{align*} B(x,y) & =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\\ & =\frac{x-1}{x+y-1}\frac{\Gamma(x-1)\Gamma(y)}{\Gamma(x+y-1)}\\ & =\frac{x-1}{x+y-1}B(x-1,y) \end{align*}

(5)

\begin{align*} B(x,y) & =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\left(\frac{x}{x+y}+\frac{y}{x+y}\right)\\ & =\frac{\Gamma(x+1)\Gamma(y)}{\Gamma(x+y+1)}+\frac{\Gamma(x)\Gamma(y+1)}{\Gamma(x+y+1)}\\ & =B(x+1,y)+B(x,y+1) \end{align*}

(5)-2

\begin{align*} B(x,y) & =\int_{0}^{1}(t+(1-t))t^{x-1}(1-t)^{y}dt\\ & =\int_{0}^{1}t^{x}(1-t)^{y}dt+\int_{0}^{1}t^{x}(1-t)^{y+1}dt\\ & =B(x+1,y)+B(x,y+1) \end{align*}

(6)

\begin{align*} B(x,y) & =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\\ & =\frac{\Gamma(x)}{Q(y,x)} \end{align*}