Carlitz exponential

In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis.

The completion C∞ of an algebraic closure of the field Fq((T−1)) of formal Laurent series in T−1 will be useful.

First we need analogues to the factorials, which appear in the definition of the usual exponential function.

Using this we define the Carlitz exponential eC:C∞ → C∞ by the convergent sum The Carlitz exponential satisfies the functional equation where we may view

By the universal property of polynomial rings in one variable this extends to a ring homomorphism ψ:Fq[T]→C∞{τ}, defining a Drinfeld Fq[T]-module over C∞{τ}.