  This paper concerns the construction of $su(r+1)$ Barut--Girardello coherent
states in term of generalized Grassmann variables. We first introduce a
generalized Weyl-Heisenberg algebra ${\cal A}(r)$ ($r \geq 1$) generated by $r$
pairs of creation and annihilation operators. This algebra provides a useful
framework to describe qubit and qukit ($k$-level) systems. It includes the
usual Weyl-Heisenberg and $su(2)$ algebras. We investigate the corresponding
Fock representation space. The generalized Grassmann variables are introduced
as variables spanning the Fock--Bargmann space associated with the algebra
${\cal A}(r)$. The Barut--Girardello coherent states for $su(r+1)$ algebras are
explicitly derived and their over--completion properties are discussed.
