ヴァンデルモンドの恒等式みたいな畳み込みは、多項係数でも成り立つのか試したらうまくいきました。証明も出来たので見てください。
まず下降冪多項定理を証明します。(x)nを下降冪とします。
(x0+x1+x2+…+xr)n+1=(x0+x1+x2+…+xr−n)(x0+x1+x2+…+xr)n=∑y0+y1+y2+…+yr=n((x0−y0)+(x1−y1)+(x2−y2)+…+(xr−yr))(ny0y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)((x0−y0)+(x1−y1)+(x2−y2)+…+(xr−yr))(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)((x0−y0)(x0)y0(x1)y1(x2)y2…(xr)yr+(x1−y1)(x0)y0(x1)y1(x2)y2…(xr)yr+(x2−y2)(x0)y0(x1)y1(x2)y2…(xr)yr+…+(xr−yr)(x0)y0(x1)y1(x2)y2…(xr)yr)=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)((x0)y0+1(x1)y1(x2)y2…(xr)yr+(x0)y0(x1)y1+1(x2)y2…(xr)yr+(x0)y0(x1)y1(x2)y2+1…(xr)yr+…+(x0)y0(x1)y1(x2)y2…(xr)yr+1)=∑(y0+1)+y1+y2+…+yr=n+1(n(y0−1)y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr+∑y0+(y1+1)+y2+…+yr=n+1(ny0(y1−1)y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr+∑y0+y1+(y2+1)+…+yr=n+1(ny0y1(y2−1)…yr)(x0)y0(x1)y1(x2)y2…(xr)yr…+∑y0+y1+y2+…+(yr+1)=n+1(ny0y1y2…(yr−1))(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n+1((n(y0−1)y1y2…yr) +(ny0(y1−1)y2…yr)+ (ny0y1(y2−1)…yr)+… +(ny0y1y2…(yr−1)))(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n+1(n+1y0y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr
これを踏まえてさらに進みます。
(x0+x1+x2+…+xr)n=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr(x0+x1+x2+…+xr)!(x0+x1+x2+…+xr)−n!=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)x0!x0−y0!x1!x1−y1!x2!x2−y2!…xr!xr−yr!(x0+x1+x2+…+xr)!(x0+x1+x2+…+xr)−n!n!=∑y0+y1+y2+…+yr=nx0!x0−y0!y0!x1!x1−y1!y1!x2!x2−y2!y2!…xr!xr−yr!yr!
(x0x1x2…xrn)==∑y0+y1+y2+…+yr=n(x0y0) (x1y1)(x2y2)…(xryr)
(x0+x1+x2+…+xr)n=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr(x0+x1+x2+…+xr)!(x0+x1+x2+…+xr)−n!=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)x0!x0−y0!x1!x1−y1!x2!x2−y2!…xr!xr−yr!(x0+x1+x2+…+xr)!x0!x1!x2!…xr!=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)(x0+x1+x2+…+xr)−n!(x0−y0)!(x1−y1)!(x2−y2)!…(xr−yr)!
=(x0+x1+x2+…+xr)=rとすると
(rx0x1x2…xr) ==∑y0+y1+y2+…+yr=n(ny0y1y2…yr) (r−n(x0−y0)(x1−y1)(x2−y2)…(xr−yr))
(x)nを上昇冪とします。
(x0+x1+x2+…+xr)n+1=(x0+x1+x2+…+xr+n)(x0+x1+x2+…+xr)n=∑y0+y1+y2+…+yr=n((x0+y0)+(x1+y1)+(x2+y2)+…+(xr+yr))(ny0y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)((x0+y0)+(x1+y1)+(x2+y2)+…+(xr+yr))(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)((x0+y0)(x0)y0(x1)y1(x2)y2…(xr)yr+(x1+y1)(x0)y0(x1)y1(x2)y2…(xr)yr+(x2+y2)(x0)y0(x1)y1(x2)y2…(xr)yr+…+(xr+yr)(x0)y0(x1)y1(x2)y2…(xr)yr)=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)((x0)y0+1(x1)y1(x2)y2…(xr)yr+(x0)y0(x1)y1+1(x2)y2…(xr)yr+(x0)y0(x1)y1(x2)y2+1…(xr)yr+…+(x0)y0(x1)y1(x2)y2…(xr)yr+1)=∑(y0+1)+y1+y2+…+yr=n+1(n(y0−1)y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr+∑y0+(y1+1)+y2+…+yr=n+1(ny0(y1−1)y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr+∑y0+y1+(y2+1)+…+yr=n+1(ny0y1(y2−1)…yr)(x0)y0(x1)y1(x2)y2…(xr)yr…+∑y0+y1+y2+…+(yr+1)=n+1(ny0y1y2…(yr−1))(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n+1((n(y0−1)y1y2…yr) +(ny0(y1−1)y2…yr)+ (ny0y1(y2−1)…yr)+… +(ny0y1y2…(yr−1)))(x0)y0(x1)y1(x2)y2…(xr)yr=∑y0+y1+y2+…+yr=n+1(n+1y0y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr
(x0+x1+x2+…+xr)n=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)(x0)y0(x1)y1(x2)y2…(xr)yr(x0+x1+x2+…+xr)+n−1!(x0+x1+x2+…+xr)−1!=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)(x0+y0−1)!x0−1!(x1+y1−1)!x1−1!(x2+y2−1)!x2−1!…(xr+yr−1)!xr−1!(x0+x1+x2+…+xr)+n−1!(x0+x1+x2+…+xr)−1!=∑y0+y1+y2+…+yr=n(ny0y1y2…yr)(x0+y0−1)!x0−1!(x1+y1−1)!x1−1!(x2+y2−1)!x2−1!…(xr+yr−1)!xr−1!(x0+x1+x2+…+xr)+n−1!(x0+x1+x2+…+xr)−1!n!=∑y0+y1+y2+…+yr=n(x0+y0−1)!x0−1!y0!(x1+y1−1)!x1−1!y1!(x2+y2−1)!x2−1!y2!…(xr+yr−1)!xr−1!yr!
(x0+x1+x2+…+xr+n−1n)==∑y0+y1+y2+…+yr=n(x0+y0−1y0) (x1+y1−1y1)(x2+y2−1y2)…(xr+yr−1yr)
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