Question Number 215540 by MrGaster last updated on 10/Jan/25
$$\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} \leq\mathrm{1}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} \right)^{{m}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \right)^{\mathrm{2}{k}} \underset{\mathrm{1}\leq{i}\leq{n}} {\prod}{dx}_{{i}} \\ $$
Answered by MrGaster last updated on 10/Jan/25
![∫_0 ^1 r^(2m) dr∫_Σ_(1≤i≤x) x_i ^2 =r^2 (Σ_(1≤i≤n) a_i x_i )^(2k) ds ∫_0 ^1 r^(2m+n+1) dr∫_(Σ_(1≤i≤n) x_i ^2 =1) (Σ_(1≤i≤n) a_i x_i )^(2k) ds lf n=2v+3∈2N+3,In order to avoid the combinationf o n and v below the score. I=(1/(2m+n+2))∫_(Σ_(1≤i≤n) x_i ^2 =1) (Σ_(1≤i≤n) a_i x_i )^(2k) ds =(1/(2m+n+2)) ((2^(v+2) π^(v+1) )/((2k+1)!(2k+3)…(2k+2v+1)))Δ^k (Σ_(1≤i≤n) a_i x_i )^(2k) Δ^k (Σ_(1≤i≤2v+3) a_i x_i )^(2k) =[(∂^2 /∂x_1 ^2 )+(∂^2 /∂x_2 ^2 )+…+(∂^2 /∂x_n ^2 )]^k e^((1/a_1 )Σ_(2≤i≤n) a_i x_i (∂/∂x_i )) (a_1 x_1 ) =[(∂^2 /∂x_1 ^2 )+(∂^2 /∂x_2 ^2 )+…+(∂^2 /∂x_n ^2 )]^k e^((1/a_1 )Σ_(2≤i≤n) a_i x_i (∂/∂x_1 )) (a_1 x_1 )^(2k) =[(∂^2 /∂x_1 ^2 )+(∂^2 /∂x_2 ^2 )+…+(∂^2 /∂x_n ^2 )]^k e^((1/a)Σ_(2≤i≤n) a_i x_i (∂/∂x_1 ^(2k) )x_1 ^(2k) ) =e^((1/a)Σ_(2≤i≤n) a_i x_i (∂/∂x_i )) (Σ_(1≤i≤n) a_k ^2 )^k (∂^(2k) /∂x_1 ^(2k) )x_1 ^(2k) =(Σ_(1≤i≤n) a_i ^2 )(2k)! I=((2^(v+2) π^(v+1) (Σ_(1≤i≤n) a_i ^2 )^k )/((2m+n+2)(2k+1)(2k+3)…(2k+2v+1))) lf n=2v+2∈2N+2 I=(1/(2m+n+2))∫_(Σ_(1≤i≤n) x_i ^2 =1) (Σ_(1≤i≤n) a_i +x_i )^(2k) ds=(1/(2m+n+2)) ((2π^(v+1) )/(4^k k!(v+k)))Δ^k (Σ_(1≤i≤n) a_i x_i )^(2k) =((2π^(v+1) (2k)!(Σ_(1≤i≤n) a_i ^2 )^k )/((2m+n+2)4^k k!(v+k)!))](https://www.tinkutara.com/question/Q215565.png)
$$\int_{\mathrm{0}} ^{\mathrm{1}} {r}^{\mathrm{2}{m}} {dr}\int_{\underset{\mathrm{1}\leq{i}\leq{x}} {\sum}} {x}_{{i}} ^{\mathrm{2}} ={r}^{\mathrm{2}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \right)^{\mathrm{2}{k}} {ds} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {r}^{\mathrm{2}{m}+{n}+\mathrm{1}} {dr}\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} =\mathrm{1}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \right)^{\mathrm{2}{k}} {ds} \\ $$$$\mathrm{lf}\:{n}=\mathrm{2}{v}+\mathrm{3}\in\mathrm{2}\mathbb{N}+\mathrm{3},\mathrm{In}\:\mathrm{order}\:\mathrm{to}\:\mathrm{avoid}\:\mathrm{the}\:\mathrm{combinationf} \\ $$$$\mathrm{o}\:{n}\:\mathrm{and}\:{v}\:\mathrm{below}\:\mathrm{the}\:\mathrm{score}. \\ $$$${I}=\frac{\mathrm{1}}{\mathrm{2}{m}+{n}+\mathrm{2}}\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} =\mathrm{1}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \right)^{\mathrm{2}{k}} {ds} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{m}+{n}+\mathrm{2}}\:\frac{\mathrm{2}^{{v}+\mathrm{2}} \pi^{{v}+\mathrm{1}} }{\left(\mathrm{2}{k}+\mathrm{1}\right)!\left(\mathrm{2}{k}+\mathrm{3}\right)\ldots\left(\mathrm{2}{k}+\mathrm{2}{v}+\mathrm{1}\right)}\Delta^{{k}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \right)^{\mathrm{2}{k}} \\ $$$$\Delta^{{k}} \left(\underset{\mathrm{1}\leq{i}\leq\mathrm{2}{v}+\mathrm{3}} {\sum}{a}_{{i}} {x}_{{i}} \right)^{\mathrm{2}{k}} =\left[\frac{\partial^{\mathrm{2}} }{\partial{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} }{\partial{x}_{\mathrm{2}} ^{\mathrm{2}} }+\ldots+\frac{\partial^{\mathrm{2}} }{\partial{x}_{{n}} ^{\mathrm{2}} }\right]^{{k}} {e}^{\frac{\mathrm{1}}{{a}_{\mathrm{1}} }\underset{\mathrm{2}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \frac{\partial}{\partial{x}_{{i}} }} \left({a}_{\mathrm{1}} {x}_{\mathrm{1}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left[\frac{\partial^{\mathrm{2}} }{\partial{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} }{\partial{x}_{\mathrm{2}} ^{\mathrm{2}} }+\ldots+\frac{\partial^{\mathrm{2}} }{\partial{x}_{{n}} ^{\mathrm{2}} }\right]^{{k}} {e}^{\frac{\mathrm{1}}{{a}_{\mathrm{1}} }\underset{\mathrm{2}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \frac{\partial}{\partial{x}_{\mathrm{1}} }} \left({a}_{\mathrm{1}} {x}_{\mathrm{1}} \right)^{\mathrm{2}{k}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left[\frac{\partial^{\mathrm{2}} }{\partial{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} }{\partial{x}_{\mathrm{2}} ^{\mathrm{2}} }+\ldots+\frac{\partial^{\mathrm{2}} }{\partial{x}_{{n}} ^{\mathrm{2}} }\right]^{{k}} {e}^{\frac{\mathrm{1}}{{a}}\underset{\mathrm{2}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \frac{\partial}{\partial{x}_{\mathrm{1}} ^{\mathrm{2}{k}} }{x}_{\mathrm{1}} ^{\mathrm{2}{k}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={e}^{\frac{\mathrm{1}}{{a}}\underset{\mathrm{2}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \frac{\partial}{\partial{x}_{{i}} }} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{k}} ^{\mathrm{2}} \right)^{{k}} \frac{\partial^{\mathrm{2}{k}} }{\partial{x}_{\mathrm{1}} ^{\mathrm{2}{k}} }{x}_{\mathrm{1}} ^{\mathrm{2}{k}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} ^{\mathrm{2}} \right)\left(\mathrm{2}{k}\right)! \\ $$$${I}=\frac{\mathrm{2}^{{v}+\mathrm{2}} \pi^{{v}+\mathrm{1}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} ^{\mathrm{2}} \right)^{{k}} }{\left(\mathrm{2}{m}+{n}+\mathrm{2}\right)\left(\mathrm{2}{k}+\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{3}\right)\ldots\left(\mathrm{2}{k}+\mathrm{2}{v}+\mathrm{1}\right)} \\ $$$$\mathrm{lf}\:{n}=\mathrm{2}{v}+\mathrm{2}\in\mathrm{2}\mathbb{N}+\mathrm{2} \\ $$$${I}=\frac{\mathrm{1}}{\mathrm{2}{m}+{n}+\mathrm{2}}\int_{\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{x}_{{i}} ^{\mathrm{2}} =\mathrm{1}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} +{x}_{{i}} \right)^{\mathrm{2}{k}} {ds}=\frac{\mathrm{1}}{\mathrm{2}{m}+{n}+\mathrm{2}}\:\frac{\mathrm{2}\pi^{{v}+\mathrm{1}} }{\mathrm{4}^{{k}} {k}!\left({v}+{k}\right)}\Delta^{{k}} \left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} {x}_{{i}} \right)^{\mathrm{2}{k}} \\ $$$$=\frac{\mathrm{2}\pi^{{v}+\mathrm{1}} \left(\mathrm{2}{k}\right)!\left(\underset{\mathrm{1}\leq{i}\leq{n}} {\sum}{a}_{{i}} ^{\mathrm{2}} \right)^{{k}} }{\left(\mathrm{2}{m}+{n}+\mathrm{2}\right)\mathrm{4}^{{k}} {k}!\left({v}+{k}\right)!} \\ $$