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I need to calculate the following definite integral of Gumbel functions:

$$\int_{-\infty}^{+\infty}e^{\frac{x-\alpha}{\beta}}e^{-e^{\frac{x-\alpha}{\beta}}}e^{-e^{-\frac{x-\gamma}{\delta}}}dx,$$

given real parameters $\alpha$, $\beta$, $\gamma$, $\delta$. In particular $\beta,\delta > 0$. I tried to apply the following change of variable:

$$z=e^{\frac{x-\alpha}{\beta}},$$

so that $dz=\frac{1}{\beta}zdx$ and $e^{-\frac{x-\gamma}{\delta}}=az^{b}$, where $a=e^{\frac{\gamma-\alpha}{\delta}}$ and $b=-\frac{\beta}{\delta}$. In this way, the integral can be written as follows:

$$\beta\int_{0}^{+\infty}e^{-z}e^{-az^{b}}dz,$$

where $b<0$. I've tried to calculate this integral by expanding one of the exponential functions in a Taylor series, for example:

$$\beta\sum_{n=0}^{+\infty}\frac{\left(-a\right)^{n}}{n!}\int_{0}^{+\infty}z^{nb}e^{-z}dz.$$

Each integral in the series looks like a Gamma function, but unfortunately they diverge, since $b<0$. According to WolframAlpha, for some special values of the parameter $b$, this integral equals a very complicated Meijer G-function. Is there a closed-form formula or a convergent series expansion for this integral (or for the original one with the Gumbel functions), given arbitrary values of the parameters $a$, $b$? Thanks in advance for any help you can provide.

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The M-Wright function can be written as $$ W_{\lambda,\mu} = \sum_{n=0}^\infty \frac{z^n}{n!\,\Gamma(\lambda\,n+\mu)}, \quad \lambda>-1 .$$ It is clear the last expression can be put in this form, with the use of the Gauss formula for the $\Gamma$-function. I'm not very familiar with this special function, but it has been around from the 1930s and has a searchable literature on the internet.

EDIT 10/31/2018: user2983638 points out that b<0 in the particular application and therefore the integration is not allowed. It is possible that the b<0 condition could be relaxed, and then perhaps the M-Wright function can be analytically continued?

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  • $\begingroup$ The integrals in the last expression that I wrote look like a Gamma function, but they diverge (therefore they are not Gamma functions). I don't think you can use the Gauss multiplication formula in this case. Am I wrong? $\endgroup$ – user2983638 Oct 31 '18 at 8:55
  • $\begingroup$ According to this link math.stackexchange.com/questions/204181/…, when $b>0$ the integral corresponds to a Fox-Wright function, but unfortunately $b<0$ in our case. $\endgroup$ – user2983638 Oct 31 '18 at 9:01

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