( a ) 0 := 1 β and β ( a ) k := β n = 1 k ( a + n - 1 ) , k = 1 , 2 , 3 , β¦ formulae-sequence assign Pochhammer-symbol π 0 1 and formulae-sequence assign Pochhammer-symbol π π superscript subscript product π 1 π π π 1 π 1 2 3 β¦ {\displaystyle{\displaystyle{\displaystyle{}{\left(a\right)_{0}}:=1\quad% \textrm{and}\quad{\left(a\right)_{k}}:=\prod_{n=1}^{k}(a+n-1),\quad k=1,2,3,% \ldots}}} {\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 } ( 1 ) n = n ! Pochhammer-symbol 1 π π {\displaystyle{\displaystyle{\displaystyle{\left(1\right)_{n}}=n!}}} {\displaystyle \pochhammer{1}{n}=n! }
\binomial β’ Ξ± β’ Ξ² := Ξ β‘ ( Ξ± + 1 ) Ξ β‘ ( Ξ² + 1 ) β’ Ξ β‘ ( Ξ± - Ξ² + 1 ) assign \binomial πΌ π½ Euler-Gamma πΌ 1 Euler-Gamma π½ 1 Euler-Gamma πΌ π½ 1 {\displaystyle{\displaystyle{\displaystyle{}\binomial{\alpha}{\beta}:=\frac{% \Gamma\left(\alpha+1\right)}{\Gamma\left(\beta+1\right)\Gamma\left(\alpha-% \beta+1\right)}}}} {\displaystyle \index{Binomial coefficient} \binomial{\alpha}{\beta}:=\frac{\EulerGamma@{\alpha+1}}{\EulerGamma@{\beta+1}\EulerGamma@{\alpha-\beta+1}} } \binomial β’ Ξ± β’ k := ( - Ξ± ) k k ! β’ ( - 1 ) k assign \binomial πΌ π Pochhammer-symbol πΌ π π superscript 1 π {\displaystyle{\displaystyle{\displaystyle\binomial{\alpha}{k}:=\frac{{\left(-% \alpha\right)_{k}}}{k!}(-1)^{k}}}} {\displaystyle \binomial{\alpha}{k}:=\frac{\pochhammer{-\alpha}{k}}{k!}(-1)^k }
\binomial β’ n β’ k := n ! k ! β’ ( n - k ) ! , k = 0 , 1 , 2 , β¦ , n formulae-sequence assign \binomial π π π π π π π 0 1 2 β¦ π {\displaystyle{\displaystyle{\displaystyle\binomial{n}{k}:=\frac{n!}{k!\,(n-k)% !},\quad k=0,1,2,\ldots,n}}} {\displaystyle \binomial{n}{k}:=\frac{n!}{k!\,(n-k)!},\quad k=0,1,2,\ldots,n }
\binomial β’ 2 β’ n β’ n = ( 1 2 ) n n ! β’ 4 n \binomial 2 π π Pochhammer-symbol 1 2 π π superscript 4 π {\displaystyle{\displaystyle{\displaystyle\binomial{2n}{n}=\frac{{\left(\frac{% 1}{2}\right)_{n}}}{n!}4^{n}}}} {\displaystyle \binomial{2n}{n}=\frac{\pochhammer{\frac{1}{2}}{n}}{n!}4^n }