阶乘幂

定义

上升阶乘幂

${\displaystyle (x)^{(n)}=x(x+1)(x+2)\cdots (x+n-1)={\frac {(x+n-1)!}{(x-1)!}}}$

下降阶乘幂

${\displaystyle (x)_{n}=x(x-1)(x-2)\cdots (x-n+1)={\frac {x!}{(x-n)!}}}$

两者的关系

${\displaystyle (-x)^{(n)}=(-1)^{n}(x)_{n}\ }$

${\displaystyle (1)^{(n)}=(n)_{n}=n!\ }$

其他常用符号

${\displaystyle x^{\overline {n}}={\frac {(x+n-1)!}{(x-1)!}}}$

${\displaystyle x^{\underline {n}}={\frac {x!}{(x-n)!}}}$

${\displaystyle [f(x)]^{k/h}=f(x)\cdot f(x+h)\cdot f(x+2h)\cdots f(x+(k-1)h)}$

${\displaystyle [f(x)]^{k/-h}=f(x)\cdot f(x-h)\cdot f(x-2h)\cdots f(x-(k-1)h)}$

属性

二项式属性

${\displaystyle {\frac {(x)^{(n)}}{n!}}={x+n-1 \choose n}\quad {\mbox{and}}\quad {\frac {(x)_{n}}{n!}}={x \choose n}.}$

${\displaystyle (a+b)^{(n)}=\sum _{j=0}^{n}{n \choose j}(a)^{(n-j)}(b)^{(j)}}$
${\displaystyle (a+b)_{n}=\sum _{j=0}^{n}{n \choose j}(a)_{n-j}(b)_{j}}$

${\displaystyle x^{\underline {m}}x^{\underline {n}}=\sum _{k=0}^{m}{m \choose k}{n \choose k}k!\,x^{\underline {m+n-k}}}$

实数幂

${\displaystyle (x)^{(n)}={\frac {\Gamma (x+n)}{\Gamma (x)}}}$

${\displaystyle (x)_{n}={\frac {\Gamma (x+1)}{\Gamma (x-n+1)}}}$

阶乘幂与亚微积分

${\displaystyle \Delta x^{\underline {k}}=kx^{\underline {k-1}}\,}$

${\displaystyle \partial {x^{k}}/\partial x=kx^{k-1}\,}$

参考文献

• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren E., Concrete Mathematics: A Foundation for Computer Science, 1988, ISBN 0-201-14236-8.
• Olver, Peter J., Classical Invariant Theory, Cambridge University Press, 1999, ISBN 0521558212.