  We introduce and study noncommutative extensions of the Fourier transform and
its logarithm to the algebra of functions on the free semigroup FS(2) on two
generators with the convolution multiplication. These extensions are new types
of moment and cumulant generating functions, respectively, the latter
corresponding to the cumulants which are additive under the so-called filtered
convolution on the free *-algebra on two generators. This algebra plays the
role of a ``noncommutative plane'' built on the ``classical real line'' and the
``boolean real line''. The restrictions of the cumulant generating function to
the commutative subsemigroups generated by single generators give the logarithm
of the Fourier transform and the K-transform in the boolean case, respectively.
In turn, the moment generating function is a ``semigroup interpolation''
between the Fourier transform and the Cauchy transform. Using suitable weight
function $W$ on the semigroup, both generating functions become elements of the
Banach algebra $l^{1}(FS(2),W)$. The main results of the paper are based on the
new combinatorics developed for the cumulants, the Moebius function and the
moment-cumulant formulas.
