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Can someone give me an idea, how to derive the integration? $$\int_0^{\infty}e^{-e^{t}}t^{k}\mathrm{d}t~\text{for}~k\in \mathbb N^{+}.$$ I found that the function may has a connection with the Gamma function or the Meijer G-function, but I can not know haow to solve this problem.

Liyanpeng
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    Its solution can be written as MeijerG functions: https://en.wikipedia.org/wiki/Meijer_G-function – Z Ahmed Dec 31 '20 at 07:47
  • Thank you for your helpful hints. I have know there are some connections between the MeijerG function and the integration above. But I am interested in the detailed calculation and the relation to Gamma function. – Liyanpeng Jan 06 '21 at 12:15

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$$I_k=\int_0^{\infty}e^{-e^{t}}\,t^{k}\,dt$$ As far as I know, there is no other representation of this integral beside the Meijer G hypergeometric function (except for $k=0$) provided $k$ to be an integer (if $k$ is not an integer, I do not know any formula).

The patterm is very simple $$I_1=G_{2,3}^{3,0}\left(1\left| \begin{array}{c} 1,1 \\ 0,0,0 \end{array} \right.\right)$$

$$I_2=2 \,G_{3,4}^{4,0}\left(1\left| \begin{array}{c} 1,1,1 \\ 0,0,0,0 \end{array} \right.\right)$$

$$I_3=6 \,G_{4,5}^{5,0}\left(1\left| \begin{array}{c} 1,1,1,1 \\ 0,0,0,0,0 \end{array} \right.\right)$$ that is to say $$I_k=k! \,G_{k+1,k+2}^{k+2,0}\left(1\left| \begin{array}{c} 1,1,1,1,\cdots \\ 0,0,0,0,0,\cdots \end{array} \right.\right)$$

If you want a table of numerical values, let me know and I shall edit.

Claude Leibovici
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  • Thank you for your helpful answer. Is there a detailed process of the result? Or can we derive it by induction? I am also interested in its relation to the Gamma function, could you please give some hints or satements about it? Thank you again. – Liyanpeng Jan 06 '21 at 12:20