On the generalized Carlitz module

The aim of this note is to gather formal similarities between two apparently different functions; {\em Euler's function} $Γ$ and {\em Anderson-Thakur function} $ω$. We discuss these similarities in the framework of the {\em generalized Carlitz's module}, a common structure which can be helpful in framing the theories of both functions. We further analyze several noticeable differences while investigating the {\em exponential functions} associated to these structures.

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