$$B_x\left(a,b\right)=\frac{{\left(a+b\right)}^{\stackrel{-}{n}}}{a^{\stackrel{-}{n}}}{B}_x\left(a+n,b\right)+\frac{x^a{\left(1-x\right)}^b}{a}\sum _{j=0}^{n-1}\frac{{\left(a+b\right)}^{\stackrel{-}{j}}}{{\left(a+1\right)}^{\stackrel{-}{j}}}{x}^j$$