  A difference operator realization of quantum deformed oscillator algebra
$H_q(1)$ with a Casimir operator freedom is introduced. We show that this
$H_q(1)$ have a nonlinear mapping to the deformed quantum su(2) which was
introduced by Fujikawa et al. We also examine the cyclic representation
obtained by this difference operator realization and the possibility to analyze
a Bloch electron problem by $H_q(1)$.
