I could not understand that whyGAMMA function is used for interpolation? Isn't it wrong to say that that factorial of 4.3 is gamma 3.3?
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The gamma function (no need to write that in caps) is used because it (with an offset of $1$) agrees with the factorial function and gives good results. There are plenty of questions here that explains it, try searching for some of those. – Henrik supports the community Apr 06 '15 at 13:21
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Like you hint, there are many functions on the positive real numbers that interpolate the map $n \mapsto (n - 1)!$ on the set of positive integers $n$. The Bohr-Mollerup Theorem asserts that the Gamma function $\Gamma$ (or more precisely, its restriction $\Gamma\vert_{\mathbb{R}_+}$ to positive real numbers) is the unique such interpolation satisfying an additional natural condition, namely, log-convexity.
One can then uniquely extend the function by analytic continuation to the usual Gamma function $\Gamma$ defined on all of the complex plane less isolated singularities.
As for the notation $(4.3)!$, it certainly looks peculiar to my eye, but there is no other likely interpretation besides the one furnished by $\Gamma$, and one does occasionally see $!$ applied to half-integers, e.g., $\Gamma(\frac{1}{2}) = \sqrt{\pi}$.
Travis Willse
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