不完全ベータ関数の漸化式
不完全ベータ関数\(B\left(z;\alpha,\beta\right)\)は次の漸化式を満たす。

(1)

\[ B\left(z;\alpha+1,\beta\right)=\frac{1}{\alpha+\beta}\left(\alpha B\left(z;\alpha,\beta\right)-z^{\alpha}\left(1-z\right)^{\beta}\right) \]

(2)

\[ B\left(z;\alpha,\beta+1\right)=\frac{1}{\alpha+\beta}\left(\beta B\left(z;\alpha,\beta\right)+z^{\alpha}\left(1-z\right)^{\beta}\right) \]

(1)

\begin{align*} B\left(z;\alpha+1,\beta\right) & =\int_{0}^{z}t^{\alpha}\left(1-t\right)^{\beta-1}dt\\ & =-\frac{1}{\beta}\left[t^{\alpha}\left(1-t\right)^{\beta}\right]_{0}^{z}+\frac{\alpha}{\beta}\int_{0}^{z}t^{\alpha-1}\left(1-t\right)^{\beta}dt\\ & =-\frac{1}{\beta}z^{\alpha}\left(1-z\right)^{\beta}+\frac{\alpha}{\beta}\int_{0}^{z}t^{\alpha-1}\left(1-t\right)\left(1-t\right)^{\beta-1}dt\\ & =-\frac{1}{\beta}z^{\alpha}\left(1-z\right)^{\beta}+\frac{\alpha}{\beta}\left(\int_{0}^{z}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt-\int_{0}^{z}t^{\alpha}\left(1-t\right)^{\beta-1}dt\right)\\ & =-\frac{1}{\beta}z^{\alpha}\left(1-z\right)^{\beta}+\frac{\alpha}{\beta}B\left(z;\alpha,\beta\right)-\frac{\alpha}{\beta}\LHS\\ & =\frac{1}{\alpha+\beta}\left(\alpha B\left(z;\alpha,\beta\right)-z^{\alpha}\left(1-z\right)^{\beta}\right) \end{align*}

(2)

\begin{align*} B\left(z;\alpha,\beta+1\right) & =\int_{0}^{z}t^{\alpha-1}\left(1-t\right)^{\beta}dt\\ & =\frac{1}{\alpha}\left[t^{\alpha}\left(1-t\right)^{\beta}\right]_{0}^{z}+\frac{\beta}{\alpha}\int_{0}^{z}t^{\alpha}\left(1-t\right)^{\beta-1}dt\\ & =\frac{1}{\alpha}z^{\alpha}\left(1-z\right)^{\beta}-\frac{\beta}{\alpha}\int_{0}^{z}t^{\alpha-1}\left(1-t-1\right)\left(1-t\right)^{\beta-1}dt\\ & =\frac{1}{\alpha}z^{\alpha}\left(1-z\right)^{\beta}-\frac{\beta}{\alpha}\left(\int_{0}^{z}t^{\alpha-1}\left(1-t\right)^{\beta}dt-\int_{0}^{z}t^{\alpha-1}\left(1-t\right)^{\beta-1}dt\right)\\ & =\frac{1}{\alpha}z^{\alpha}\left(1-z\right)^{\beta}-\frac{\beta}{\alpha}\LHS+\frac{\beta}{\alpha}B\left(z;\alpha,\beta\right)\\ & =\frac{1}{\alpha+\beta}\left(\beta B\left(z;\alpha,\beta\right)+z^{\alpha}\left(1-z\right)^{\beta}\right) \end{align*}