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.
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