多項定理
より、 = ( x 1 + ⋯ + x m ) n + 1 {\displaystyle {\hphantom {=}}\;(x_{1}+\cdots +x_{m})^{n+1}} = ( x 1 + ⋯ + x m ) ( x 1 + ⋯ + x m ) n {\displaystyle =(x_{1}+\cdots +x_{m})(x_{1}+\cdots +x_{m})^{n}} = ( x 1 + ⋯ + x m ) ∑ k 1 + ⋯ + k m = n ( n k 1 , ⋯ , k m ) x 1 k 1 ⋯ x m k m {\displaystyle =(x_{1}+\cdots +x_{m})\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n}{\dbinom {n}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}} = x 1 ∑ k 1 + ⋯ + k m = n ( n k 1 , ⋯ , k m ) x 1 k 1 ⋯ x m k m + x m ∑ k 1 + ⋯ + k m = n ( n k 1 , ⋯ , k m ) x 1 k 1 ⋯ x m k m {\displaystyle =x_{1}\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n}{\dbinom {n}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}+x_{m}\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n}{\dbinom {n}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}} = ∑ k 1 + ⋯ + k m = n ( n k 1 , ⋯ , k m ) x 1 k 1 + 1 x 2 k 2 ⋯ x m k m + ∑ k 1 + ⋯ + k m = n ( n k 1 , ⋯ , k m ) x 1 k 1 ⋯ x m − 1 k m − 1 x m k m + 1 {\displaystyle =\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n}{\dbinom {n}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}+1}{x_{2}}^{k_{2}}\cdots {x_{m}}^{k_{m}}+\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n}{\dbinom {n}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m-1}}^{k_{m-1}}{x_{m}}^{k_{m}+1}} = ∑ k 1 + ⋯ + k m = n + 1 k 1 ≥ 1 ( n k 1 , ⋯ , k m ) x 1 k 1 ⋯ x m k m + ∑ k 1 + ⋯ + k m = n k m ≥ 1 ( n k 1 , ⋯ , k m ) x 1 k 1 ⋯ x m k m {\displaystyle =\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n+1 \atop k_{1}\geq 1}{\dbinom {n}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}+\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n \atop k_{m}\geq 1}{\dbinom {n}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}} = ∑ k 1 + ⋯ + k m = n + 1 ( n k 1 − 1 , k 2 , ⋯ , k m ) x 1 k 1 ⋯ x m k m + ∑ k 1 + ⋯ + k m = n + 1 ( n k 1 , ⋯ , k m − 1 , k m − 1 ) x 1 k 1 ⋯ x m k m {\displaystyle =\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n+1}{\dbinom {n}{k_{1}-1,k_{2},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}+\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n+1}{\dbinom {n}{k_{1},\cdots ,k_{m-1},k_{m}-1}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}} ( ∵ ( n ⋯ , − 1 , ⋯ ) = 0 ) {\displaystyle \left(\because {\binom {n}{\cdots ,-1,\cdots }}=0\right)} = ∑ k 1 + ⋯ + k m = n + 1 [ ( n k 1 − 1 , k 2 , ⋯ , k m ) + ⋯ + ( n k 1 , ⋯ , k m − 1 , k m − 1 ) ] x 1 k 1 ⋯ x m k m {\displaystyle =\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n+1}\left[{\dbinom {n}{k_{1}-1,k_{2},\cdots ,k_{m}}}+\cdots +{\dbinom {n}{k_{1},\cdots ,k_{m-1},k_{m}-1}}\right]{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}} = ∑ k 1 + ⋯ + k m = n + 1 ( n + 1 k 1 , ⋯ , k m ) x 1 k 1 ⋯ x m k m {\displaystyle =\textstyle \sum \limits _{k_{1}+\cdots +k_{m}=n+1}{\dbinom {n+1}{k_{1},\cdots ,k_{m}}}{x_{1}}^{k_{1}}\cdots {x_{m}}^{k_{m}}}
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出典: フリー百科事典『ウィキペディア（Wikipedia）』
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