  The primary objective of this paper is to derive explicit formulas for rank
one and rank two Drinfeld modules over a specific domain denoted by A. This
domain corresponds to the projective line associated with an infinite place of
degree two. To achieve the goals, we construct a pair of standard Drinfeld
modules whose coefficients are in the Hilbert class field of A. We demonstrate
that the period lattice of the exponential functions corresponding to both
modules behaves similarly to the period lattice of the Carlitz module, the
standard rank one Drinfeld module defined over rational function field.
Moreover, we employ Andersons t-motive to obtain the complete family of rank
two Drinfeld modules. This family is parameterized by the invariant J =
\lambda^{q^2+1} which effectively serves as the counterpart of the j-invariant
for elliptic curves. Building upon the concepts introduced by van~der~Heiden,
particularly with regard to rank two Drinfeld modules, we are able to
reformulate the Weil pairing of Drinfeld modules of any rank using a
specialized polynomial in multiple variables known as the Weil operator. As an
illustrative example, we provide a detailed examination of a more explicit
formula for the Weil pairing and the Weil operator of rank two Drinfeld modules
over the domain A.
