A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A=P(d/dx+x)/ \ensuremath{\surd}2 , where P is the parity operator. Such A arises naturally in the q\ensuremath{\rightarrow}-1 limit for a symmetry operator of a specific self-similar potential obeying the q-Weyl algebra ${\mathit{AA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$-
q2${\mathit{A}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$A=1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit, the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states \ensuremath{\Vert}${\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}$\ensuremath{\alpha}〉, k=0,1,...,N-1, where \ensuremath{\epsilon} is a primitive Nth root of unity, ${\mathrm{\ensuremath{\epsilon}}}^{\mathit{N}}$=1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes an unusual set of coherent states of the free nonrelativistic particle, which is interpreted as a q-algebraic system without a discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural q analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.