  By resorting to the Fock--Bargmann representation, we incorporate the quantum
Weyl--Heisenberg algebra, $q$-WH, into the theory of entire analytic functions.
The $q$--WH algebra operators are realized in terms of finite difference
operators in the $z$ plane. In order to exhibit the relevance of our study,
several applications to different cases of physical interest are discussed:
squeezed states and the relation between coherent states and theta functions on
one side, lattice quantum mechanics and Bloch functions on the other, are shown
to find a deeper mathematical understanding in terms of $q$-WH. The r\^ole
played by the finite difference operators and the relevance of the lattice
structure in the completeness of the coherent states system suggest that the
quantization of the WH algebra is an essential tool in the physics of
discretized (periodic) systems.
