# The shifted factorial and binomial coefficients

## The shifted factorial and binomial coefficients

$\displaystyle {\displaystyle \index{Shifted factorial} \pochhammer{a}{0}:=1\quad\textrm{and}\quad \pochhammer{a}{k}:=\prod_{n=1}^k(a+n-1),\quad k=1,2,3,\ldots }$
$\displaystyle {\displaystyle \pochhammer{1}{n}=n! }$

Constraint(s): $\displaystyle {\displaystyle n=0,1,2,\ldots}$

$\displaystyle {\displaystyle \index{Binomial coefficient} \binomial{\alpha}{\beta}:=\frac{\EulerGamma@{\alpha+1}}{\EulerGamma@{\beta+1}\EulerGamma@{\alpha-\beta+1}} }$
$\displaystyle {\displaystyle \binomial{\alpha}{k}:=\frac{\pochhammer{-\alpha}{k}}{k!}(-1)^k }$

Constraint(s): $\displaystyle {\displaystyle k=0,1,2,\ldots}$

$\displaystyle {\displaystyle \binomial{n}{k}:=\frac{n!}{k!\,(n-k)!},\quad k=0,1,2,\ldots,n }$

Constraint(s): $\displaystyle {\displaystyle n=0,1,2,\ldots}$

$\displaystyle {\displaystyle \binomial{2n}{n}=\frac{\pochhammer{\frac{1}{2}}{n}}{n!}4^n }$

Constraint(s): $\displaystyle {\displaystyle n=0,1,2,\ldots}$