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Recently I am struglling with the nested integration of incomplete gamma function.

[tex]\int_{0}^{\infty}\int_{0}^{y}x^{\alpha-1}e^{-x/\beta}y^{\alpha-1}e^{-y/\beta}dx dy[/tex]

after integre 'x', we can get

[tex]\int_{0}^{\infty}\beta^{\alpha}\gamma(\alpha, y/\beta)y^{\alpha-1}e^{-y/\beta }dy[/tex]

I know that i can integre it directly and get the result which involves gussian hypergeometric function 2F1. But i want to use the series expansion,

[tex]\gamma(\alpha, x) = x^\alpha\sum_{k=0}^{\infty}\frac{(-x)^k}{(\alpha+k)k!}[/tex]

The result is,

[tex]\sum_{k=0}^{\infty}\frac{(-1)^k\beta^{2\alpha}\Gamma(2\alpha+k)}{(\alpha+k)k!}[/tex]

The problem is that it do not always converge, any idea?

Many thanks indeed.

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# Integartion of Incomplete Gamma Function

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