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$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
This question is the "cousin" of a closely related one.
For $x\ge0$ \begin{align} \int_0^1 (-\Wp(-\tfrac t{\e}))^x \, dt &=\frac {1+\e\,x\,(\Gamma(x+2,1)-\Gamma(x+2))}{x+1} \tag{1}\label{1} ,\\ \int_0^1 (-\Wm(-\tfrac t{\e}))^x \, dt &=\frac{\e\,x\Gamma(x+2,1)+1}{x+1} \tag{2}\label{2} ,\\ \int_0^1 (-\Wm(-\tfrac t{\e}))^x-(-\Wp(-\tfrac t{\e}))^x \, dt &=\e\,x\,\Gamma(x+1) \tag{3}\label{3} , \end{align}
where $\Wp,\ \Wm$ are the real branches of the Lambert $\W$ function.
Equation \eqref{3} also suggests an elegant definition of the gamma function in terms of the real branches of the Lambert $\W$ function as \begin{align} \Gamma(x+1) &= \frac1{\e\,x}\int_0^1 (-\Wm(-\tfrac t{\e}))^x-(-\Wp(-\tfrac t{\e}))^x \, dt \tag{4}\label{4} . \end{align}
Question: Are these relations known? Any reference?
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