• $$\large p_{nk}=n(n-1)\dots(n-k+1) \tag{2}$$

• $$\large n!=1 \cdot 2 \cdot \dots \cdot n = \prod_{k=1}^n k \tag{4}$$

• $$\large n!\approx \sqrt{2\pi n}(\frac{n}{e})^n \tag{7}$$

• $$\large \mu = \sum_{k>0}\lfloor \frac{n}{p^k} \rfloor \tag{8}$$

  例子: $1000!可由3^{498}所整除$

  $$\mu=\lfloor \frac{1000}{3} \rfloor + \lfloor \frac{1000}{9} \rfloor+\dots +\lfloor \frac{1000}{729} \rfloor \\ = 333 + 111 +37+12+4+1 = 498$$

• $$\large n!=\varGamma(n+1)=n\varGamma(n) \tag{14}$$

• $$\large \varGamma(x)=\frac{x!}{x}=\lim_{m \to \infty} \frac{m^xm!}{x(x+1)(x+2)\dots(x+m)} \tag{15}$$

• $$\large (-z)!\varGamma(z)=\frac{\pi}{\sin \pi z} \tag{16}$$

• $$\large x^{\underline{k}} = x(x-1)\dots(x-k+1)=\prod_{j=0}^{k-1}(x-j) \tag{18}$$

• $$\large x^{\overline{k}} = x(x+1)\dots(x+k-1)=\prod_{j=0}^{k-1}(x+j) \tag{19}$$

• $(2)的数p_{nk}仅仅是n^{\underline{k}}，我们有$

$$x^{\overline{k}}=(x+k-1)^{\underline{k}}=(-1)^k(-x)^{\underline{k}}\tag{20}$$

• $$x^{\underline{k}}=\frac{x!}{(x-k)!}\;,\;x^{\overline{k}}= \frac{\varGamma(x+k)}{\varGamma(x)}\tag{21}$$