  A new class of states of light is introduced that is complementary to the
well-known squeezed states. The construction is based on the general solution
of the three-term recurrence relation that arises from the saturation of the
Schr\"odinger inequality for the quadratures of a single-mode quantized
electromagnetic field. The new squeezed states are found to be linear
superpositions of the photon-number states whose coefficients are determined by
the associated Hermite polynomials. These results do not seem to have been
noticed before in the literature. As an example, the new class of squeezed
states includes superpositions characterized by odd-photon number states only,
so they represent the counterpart of the prototypical squeezed-vacuum state
which consists entirely of even-photon number states.
