We work out in detail the Drinfeld module over the ring

The example in question is one of the four examples that come from quadratic imaginary fields with class number  and rank one.

We develop specific formulas for the coefficients  and  of the exponential and logarithmic functions and relate them with the product  of all monic elements of A of degree k. On the Carlitz module,  and  coincide, but this is not true for general Drinfeld modules. On this example, we obtain a formula relating both invariants. We prove also using elementary methods a theorem due to Thakur that relate two different combinatorial symbols important in the analysis of solitons.

Drinfeld module, exponential and logarithmic functions.