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I have heard that factorials of negative numbers and fractions exist. Is there any proof to this fact and what value do they assume?

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    You cannot *prove* that the factorials of negative numbers exist, since you do not have a definition for them. You should be asking «are there sensible definitions for factorials of negative numbers and fractions?» and if he answer is affirmative —and it is— then there is nothing to be proved: you do not prove definitions. – 2017-02-13

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The Gamma Function: $\Gamma (z) = \int_{0}^{\infty} t^{z-1} e^{-t} \mathrm {d}t $ is a generalisation of the factorial to non integers.

You might just want to see here and here. Hope it helps.

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    You should also mention that the gamma function has poles at negative integers. – 2017-02-13