  In a previous paper [{\it J. Phys. A: Math. Theor.} {\bf 40} (2007) 11105],
we constructed a class of coherent states for a polynomially deformed $su(2)$
algebra. In this paper, we first prepare the discrete representations of the
nonlinearly deformed $su(1,1)$ algebra. Then we extend the previous procedure
to construct a discrete class of coherent states for a polynomial su(1,1)
algebra which contains the Barut-Girardello set and the Perelomov set of the
SU(1,1) coherent states as special cases. We also construct coherent states for
the cubic algebra related to the conditionally solvable radial oscillator
problem.
