  The intimate connection between q-deformed Heisenberg uncertainty relation
and the Jackson derivative based on q-basic numbers has been noted in the
literature. The purpose of this work is to establish this connection in a clear
and self-consistent formulation and to show explicitly how the Jackson
derivative arises naturally. We utilize a holomorphic representation to arrive
at the correct algebra to describe q-deformed bosons. We investigate the
algebra of q-fermions and point out how different it is from the theory of
q-bosons. We show that the holomorphic representation for q-fermions is indeed
feasible in the framework of the theory of generalized fermions. We also
examine several different q-algebras in the context of the modified Heisenberg
equation of motion.
