  We review the definition of hypergeometric coherent states, discussing some
representative examples. Then we study mathematical and statistical properties
of hypergeometric Schr\"odinger cat states, defined as orthonormalized
eigenstates of $k$-th powers of nonlinear $f$-oscillator annihilation
operators, with $f$ of hypergeometric type. These "$k$-hypercats" can be
written as an equally weighted superposition of hypergeometric coherent states
$|z_l\rangle, l=0,1,\dots,k-1$, with $z_l=z e^{2\pi i l/k}$ a $k$-th root of
$z^k$, and they interpolate between number and coherent states. This fact
motivates a continuous circle representation for high $k$. We also extend our
study to truncated hypergeometric functions (finite dimensional Hilbert spaces)
and a discrete exact circle representation is provided. We also show how to
generate $k$-hypercats by amplitude dispersion in a Kerr medium and analyze
their generalized Husimi $Q$-function in the super- and sub-Poissonian cases at
different fractions of the revival time.
