I have an Gauss sum, which I have to colculate I have heard that it has an analogues form with the Gumma function, but couldn't find its formular shape. It would be so nice of you to help me and write the mathematical shape of the link between these two funktions. Beforehand thank you.
The connection between Gauss sums and the $p$-adic Gamma function is a very deep theorem, and given by the Gross-Koblitz formula.
Here is the original paper by Gross and Koblitz: Gauss sums and the $p$-adic $\Gamma$-function. (1979)
For some more specific cases and applications, see the book by Berndt, Evans, and Williams, "Gauss and Jacobi Sums. (1998)"