  We define two pairings relating the A-motive with the dual A-motive of an
abelian Anderson A-module. We show that specializations of these pairings give
the exponential and logarithm functions of this Anderson A-module, and we use
these specializations to give precise formulas for the coefficients of the
exponential and logarithm functions. We then use this pairing to express the
exponential and logarithm functions as evaluations of certain infinite
products. As an application of these ideas, we prove an analogue of the Mellin
tranform formula for the Riemann zeta function in the case of Carlitz zeta
values. We also give an example showing how our results apply to Carlitz
multiple zeta values.
