  We obtain and investigate the regular eigenfunctions of simple differential
operators x^r d^{r+1}/dx^{r+1}, r=1, 2, ... with the eigenvalues equal to one.
With the help of these eigenfunctions we construct a non-unitary analogue of
boson displacement operator which will be acting on the vacuum. In this way we
generate collective quantum states of the Fock space which are normalized and
equipped with the resolution of unity with the positive weight functions that
we obtain explicitly. These states are thus coherent states in the sense of
Klauder. They span the truncated Fock space without first r lowest-lying basis
states: |0>, |1>, ..., |r-1>. These states are squeezed, are sub-Poissonian in
nature and are reminiscent of photon-added states at Agarwal et al.
