Photon-added coherent states as nonlinear coherent states
  The states $|\alpha,m>$, defined as ${a^{\dagger}}^{m}|\alpha>$ up to a normalization constant and $m$ is a nonnegative integer, are shown to be the eigenstates of $f(\hat{n},m)\hat{a}$ where $f(\hat{n},m)$ is a nonlinear function of the number operator $\hat{n}$. The explicit form of $f(\hat{n},m)$ is constructed. The eigenstates of this operator for negative values of $m$ are introduced. The properties of these states are discussed and compared with those of the state $|\alpha,m >$. 
S. Sivakumar