積分のヴァンデルモンドの畳み込み(ガバ解説)

$$\begin{array}{rl}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{a}{t})(\genfrac{}{}{0ex}{}{b}{c-t})\mathrm{d}t& =((\genfrac{}{}{0ex}{}{a}{x})\ast (\genfrac{}{}{0ex}{}{b}{x}))(c)\\ & =\mathcal{F}[(1+e^{it}{)}^a(1+e^{it}{)}^b](c)\\ & =\mathcal{F}[(1+e^{it}{)}^{a+b}](c)\\ & =(\genfrac{}{}{0ex}{}{a+b}{c}).\end{array}$$

$$(\mathrm{注}\mathrm{：}\mathcal{F}[{\mathcal{F}}^{-1}[f]{\mathcal{F}}^{-1}[g]](t)=(f\ast g)(t),{\mathcal{F}}^{-1}\left[(\genfrac{}{}{0ex}{}{a}{s})\right](t)=(1+e^{it}{)}^a(a>-1,|t|<\pi ))$$

$$\begin{array}{rl}(\genfrac{}{}{0ex}{}{p+q}{r})& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{p}{t})(\genfrac{}{}{0ex}{}{q}{r-t})\mathrm{d}t\\ & =\mathrm{\Gamma }(p+1)\mathrm{\Gamma }(q+1){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{d}t}{\mathrm{\Gamma }(t+1)\mathrm{\Gamma }(p-t+1)\mathrm{\Gamma }(r-t+1)\mathrm{\Gamma }(q-r-t+1)}\\ & =\mathrm{\Gamma }(p+1)\mathrm{\Gamma }(q+1){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{d}x}{\mathrm{\Gamma }(a+x)\mathrm{\Gamma }(p-a+2-x)\mathrm{\Gamma }(r-a+2-x)\mathrm{\Gamma }(q-r+a+t)}\\ & (t+1⟼a+x)\\ \end{array}$$
$p⟼a+c-2,q⟼b+d-2,r=a+d-2$として，

$$\begin{array}{rl}\therefore {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{d}x}{\mathrm{\Gamma }(a+x)\mathrm{\Gamma }(b+x)\mathrm{\Gamma }(c-x)\mathrm{\Gamma }(d-x)}& =\frac{1}{\mathrm{\Gamma }(a+c-1)\mathrm{\Gamma }(b+d-1)}(\genfrac{}{}{0ex}{}{a+b+c+d-4}{a+d-2})\\ & =\frac{\mathrm{\Gamma }(a+b+c+d-3)}{\mathrm{\Gamma }(a+c-1)\mathrm{\Gamma }(a+d-1)\mathrm{\Gamma }(b+c-1)\mathrm{\Gamma }(b+d-1)}.\end{array}$$