$ \sum_{\beta \in {\bf N}^d} degree(\beta) ! \cdot \Phi(\beta; x) \cdot t_1^{\beta_1} \cdots t_d^{\beta_d} = \frac{1}{1 - \sum_{i=1}^n x_i \cdot t_1^{a_{1i}} \cdots t_d^{a_{di}}} $ where $degree(\beta) := u_1 + \cdots + u_n$ for any $ u = (u_1,\ldots,u_n) \in T^{-1}(\beta)$.

Let $\alpha, \gamma_1, \ldots, \gamma_{d-1}$ be integers, set $\gamma_d := - p \alpha $, $m = d-1$, and $\beta = (-\gamma_1-1, \ldots, -\gamma_{d-1}-1, -\gamma_d)^T$. Then, the hypergeometric ideal $H_A(\beta)$ annihilates the integral $ \Phi(\gamma;x) \quad = \quad \int_C \left( \sum_{i=1}^n x_i t^{a_i}\right)^{\! \alpha} t^\gamma dt_1 \cdots dt_m $ where $a_{d1} \, = \, a_{d2} \, = \, \, \cdots \, \, = \, a_{dn}=:p$.


$ \Gamma(z) \sim \sqrt{2 \pi} e^{-z} z^{z-1/2} \left( 1 + \frac{1}{12z} + \frac{1}{288z^2} - \frac{139}{51840z^3} - \cdots \right) $ $ \log \Gamma(z) \sim \frac{1}{2}(\log 2 \pi - 1) -z -\log z + \left( \frac{z}{2} + \frac{1}{4} \right) (\log z + \log(z+1)) + \frac{1}{12} \log \frac{z+1}{z} + \left( \frac{1}{180} \left(\frac{1}{z^2}-\frac{1}{(z+1)^2}\right) - \frac{1}{840} \left(\frac{1}{z^4}-\frac{1}{(z+1)^4}\right) + \cdots \right) $ $ \log \Gamma(z) \sim z \log z - z + \frac{1}{2} \log \frac{2\pi}{z} +\sum_{n=1}^\infty \frac{(-1)^{n-1} B_n}{(2n) (2n-1) z^{2n-1}} $ Stirling の近似 $\int_{-\infty}^\infty exp\left(-\frac{x^2}{2 \sigma^2}\right) = \sqrt{2 \pi \sigma^2}$ $\int_{{\bf R}^d} exp\left(-\frac{1}{2} x^T S^{-1} x \right) dx = (\sqrt{2 \pi})^d \sqrt{|S|}$