Gamma Function:
$$\Gamma(t)=\int_{0}^{\infty}x^{t-1}e^{-x}dx$$
Is it known for which values of $t$ (real or complex), the value of $\Gamma(t)$ is integer?
Are there any known specific patterns of $t$, for which the value of $\Gamma(t)$ is integer?
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1Don't think so. Even only on the positive real half line, there are many but they have no characterization that I would be aware of. – Did Apr 02 '14 at 16:47
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Hard to say. Since $\Gamma(x)=(x-1)!$ and it is continuous, it visits all integers exactly once, and between integer values of $x$.(you can get arbitrarily more integer values for the function at non-integral points between $x$ and $x+1$ by taking large enough $x$). – Guy Apr 02 '14 at 16:49
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2$\Gamma(x)$ visits all integers except zero infinitely many times if we consider that $x$ may be a negative number. – Brad Apr 02 '14 at 16:52
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Thank you @Did. I wasn't actually hoping to get an answer that would cover all integers. But are there certain patterns of $t$ for which $\Gamma(t)$ yields an integer result? – barak manos Apr 02 '14 at 16:52
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Thanks @Brad. Are there certain patterns of $t$ for which $\Gamma(t)$ yields an integer result? – barak manos Apr 02 '14 at 16:53
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Then I guess your question has to be made much more precise before some interesting answer arises. – Did Apr 02 '14 at 17:28
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Thank you @Did. Question revised. – barak manos Apr 02 '14 at 17:47