Dirichlet積分をしてみた

求める積分を$I_n({\lambda }_1,{\lambda }_2,...,{\lambda }_n;\mu )$と置く．
$n=1$の場合はベータ関数となるので明らかに成り立つ．
$$\begin{array}{rcl}{I}_1({\lambda }_1;\mu )& =& {\int }_0^1{t}_1^{{\lambda }_1-1}(1-t_1{)}^{\mu -1}d{t}_1\\ & =& \frac{\mathrm{\Gamma }({\lambda }_1)\mathrm{\Gamma }(\mu )}{\mathrm{\Gamma }({\lambda }_1+\mu )}\end{array}$$
そこで，一般の1,2,...,nまで成り立つと仮定する．そして，n+1の時を考える．
$$\begin{array}{rcl}{I}_{n+1}({\lambda }_1,{\lambda }_2,...,{\lambda }_n,{\lambda }_{n+1};\mu )& =& {\int }_0^{1}d{t}_1{t}_1^{{\lambda }_1-1}{\int }_{t_2+t_3+\cdots t_n+t_{n+1}=1-t_1\wedge t_2,t_3,...,t_n,t_{n+1}>0}d{t}_{2}d{t}_3\cdots d{t}_{n}d{t}_{n+1}{t}_2^{{\lambda }_2-1}{t}_3^{{\lambda }_3-1}\cdots t_n^{{\lambda }_n-1}{t}_{n+1}^{{\lambda }_{n+1}-1}(1-t_1-t_2-\cdots t_n-t_{n+1}{)}^{\mu -1}\\ & =& {\int }_0^{1}d{t}_1{t}_1^{{\lambda }_1-1}{\int }_0^{1-t_1}d{t}_2{t}_2^{{\lambda }_2-1}{\int }_{t_3+\cdots t_n+t_{n+1}=1-t_1-t_2\wedge t_3,...,t_n,t_{n+1}>0}d{t}_3\cdots d{t}_{n}d{t}_{n+1}{t}_3^{{\lambda }_3-1}\cdots t_n^{{\lambda }_n-1}{t}_{n+1}^{{\lambda }_{n+1}-1}(1-t_1-t_2-\cdots t_n-t_{n+1}{)}^{\mu -1}\\ & =& \cdots \\ & =& {\int }_0^{1}d{t}_1{t}_1^{{\lambda }_1-1}{\int }_0^{1-t_1}d{t}_2{t}_2^{{\lambda }_2-1}\cdots {\int }_0^{1-t_1-t_2-\cdots t_{n-1}}{t}_n^{{\lambda }_n-1}{\int }_0^{1-t_1-t_2-\cdots -t_{n-1}-t_n}d{t}_{n+1}{t}_{n+1}^{{\lambda }_{n+1}-1}(1-t_1-t_2-\cdots -t_n-t_{n+1}{)}^{\mu -1}\end{array}$$
より，$t_{n+1}=(1-t_1-t_2-\cdots -t_n)s$と置きなおすと，
$$\begin{array}{rcl}{\int }_0^{1-t_1-t_2-\cdots -t_{n-1}-t_n}d{t}_{n+1}{t}_{n+1}^{{\lambda }_{n+1}-1}(1-t_1-t_2-\cdots -t_n-t_{n+1}{)}^{\mu -1}& =& (1-t_1-t_2-\cdots -t_{n-1}-t_n{)}^{{\lambda }_{n+1}+\mu -1}{\int }_0^1{s}^{{\lambda }_{n+1}-1}(1-s{)}^{\mu -1}\\ & =& (1-t_1-t_2-\cdots -t_{n-1}-t_n{)}^{{\lambda }_{n+1}+\mu -1}\frac{\mathrm{\Gamma }({\lambda }_{n+1})\mathrm{\Gamma }(\mu )}{\mathrm{\Gamma }({\lambda }_{n+1}+\mu )}\end{array}$$
を得るので，これを先の式に代入して次式を得る．
$$\begin{array}{rcl}{I}_{n+1}({\lambda }_1,{\lambda }_2,...,{\lambda }_n,{\lambda }_{n+1};\mu )& =& {\int }_0^{1}d{t}_1{t}_1^{{\lambda }_1-1}{\int }_0^{1-t_1}d{t}_2{t}_2^{{\lambda }_2-1}\cdots {\int }_0^{1-t_1-t_2-\cdots -t_{n-1}}d{t}_n{t}_n^{{\lambda }_n-1}(1-t_1-t_2-\cdots -t_n{)}^{{\lambda }_{n+1}+\mu -1}\\ & =& I_n({\lambda }_1,{\lambda }_2,...,{\lambda }_n;{\lambda }_{n+1}+\mu )\frac{\mathrm{\Gamma }({\lambda }_{n+1})\mathrm{\Gamma }(\mu )}{\mathrm{\Gamma }({\lambda }_{n+1}\mu )}\\ & =& \frac{\mathrm{\Gamma }({\lambda }_1)\mathrm{\Gamma }({\lambda }_2)\cdots \mathrm{\Gamma }({\lambda }_n)\mathrm{\Gamma }({\lambda }_{n+1}+\mu )}{\mathrm{\Gamma }({\lambda }_1+{\lambda }_2+\cdots {\lambda }_n+{\lambda }_{n+1}+\mu )}\frac{\mathrm{\Gamma }({\lambda }_{n+1})\mathrm{\Gamma }(\mu )}{\mathrm{\Gamma }({\lambda }_{n+1}+\mu )}\\ & =& \frac{\mathrm{\Gamma }({\lambda }_1)\mathrm{\Gamma }({\lambda }_2)\cdots \mathrm{\Gamma }({\lambda }_n)\mathrm{\Gamma }({\lambda }_{n+1})\mathrm{\Gamma }(\mu )}{\mathrm{\Gamma }({\lambda }_1+{\lambda }_2+\cdots {\lambda }_n+{\lambda }_{n+1}+\mu )}\end{array}$$
よって，$n+1$の場合も成り立つことが示せたので，本定理は示された．