Talk:Stieltjes constants

A multiple integral in terms of Stieltjes constants

From this we derive (this doesn't make sense, it gives $\scriptstyle \gamma _{0}\,=\,-\infty \,$!)

$I_{1}=\int _{0}^{1}{\Bigg \{}{\frac {1}{x_{1}}}{\Bigg \}}dx_{1}=[\log x_{1}]_{0}^{1}=[0-(-\infty )]=\infty =1-\sum _{k=0}^{0}{\frac {\gamma _{k}}{k!}}=1-\gamma _{0},\,$

$I_{m+1}-I_{m}=-{\Bigg \{}\sum _{k=0}^{m}{\frac {\gamma _{k}}{k!}}-\sum _{k=0}^{m-1}{\frac {\gamma _{k}}{k!}}{\Bigg \}}=-{\frac {\gamma _{m}}{m!}},\ m\geq 1,\,$

giving

$\gamma _{m}=m!\,(I_{m}-I_{m+1})=m!\,{\Bigg \{}{\Bigg [}\int _{0}^{1}\cdots \int _{0}^{1}{\Bigg \{}{\frac {1}{x_{1}x_{2}\cdots x_{m}}}{\Bigg \}}dx_{1}dx_{2}\cdots dx_{m}{\Bigg ]}-{\Bigg [}\int _{0}^{1}\cdots \int _{0}^{1}{\Bigg \{}{\frac {1}{x_{1}x_{2}\cdots x_{m}x_{m+1}}}{\Bigg \}}dx_{1}dx_{2}\cdots dx_{m}dx_{m+1}{\Bigg ]}{\Bigg \}},\ m\geq 1.\,$

A proof(?) of

$I_{m}:=\int _{0}^{1}\cdots \int _{0}^{1}{\Bigg \{}{\frac {1}{x_{1}x_{2}\cdots x_{m}}}{\Bigg \}}dx_{1}dx_{2}\cdots dx_{m}=1-\sum _{k=0}^{m-1}{\frac {\gamma _{k}}{k!}},\ m\geq 1.\,$

is given in

Yaming Yu, A Multiple Integral in Terms of Stieltjes Constants, University of California, Irvine.

— Daniel Forgues 06:52, 4 February 2011 (UTC) — Daniel Forgues 06:30, 5 February 2011 (UTC)