This article has multiple issues. Please help improve it or discuss these issues on the talk page. ( Learn how and when to remove these template messages) This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (May 2020) ( Learn how and when to remove this message) This article may be confusing or unclear to readers. In particular, it is not explained how it is a generalization of the exponential function. Please help clarify the article. There might be a discussion about this on the talk page. (May 2020) ( Learn how and when to remove this message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2020) ( Learn how and when to remove this message) ( Learn how and when to remove this message)

In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an example of a Drinfeld module.

We work over the polynomial ring F_qT] of one variable over a finite field F_q with q elements. The completion C_∞ of an algebraic closure of the field F_q((T⁻¹)) of formal Laurent series in T⁻¹ will be useful. It is a complete and algebraically closed field.

First we need analogues to the factorials, which appear in the definition of the usual exponential function. For i > 0 we define

${\displaystyle [i]:=T^{q^{i}}-T,\,}$

${\displaystyle D_{i}:=\prod _{1\leq j\leq i}[j]^{q^{i-j}}}$

and D₀ := 1. Note that the usual factorial is inappropriate here, since n! vanishes in F_qT] unless n is smaller than the characteristic of F_qT].

Using this we define the Carlitz exponential e_C:C_∞ → C_∞ by the convergent sum

${\displaystyle e_{C}(x):=\sum _{i=0}^{\infty }{\frac {x^{q^{i}}}{D_{i}}}.}$

The Carlitz exponential satisfies the functional equation

${\displaystyle e_{C}(Tx)=Te_{C}(x)+\left(e_{C}(x)\right)^{q}=(T+\tau )e_{C}(x),\,}$

where we may view ${\displaystyle \tau }$ as the power of ${\displaystyle q}$ map or as an element of the ring ${\displaystyle F_{q}(T)\{\tau \}}$ of noncommutative polynomials. By the universal property of polynomial rings in one variable this extends to a ring homomorphism ψ:F_qT]→C_∞{τ}, defining a Drinfeld F_qT]-module over C_∞{τ}. It is called the Carlitz module.

• Goss, D. (1996). Basic structures of function field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 35. Berlin, New York: Springer-Verlag. ISBN  978-3-540-61087-8. MR  1423131.
• Thakur, Dinesh S. (2004). Function field arithmetic. New Jersey: World Scientific Publishing. ISBN  978-981-238-839-1. MR  2091265.