  We introduce the third five-parametric ordinary hypergeometric
energy-independent quantum-mechanical potential, after the Eckart and
P\"oschl-Teller potentials, which is proportional to an arbitrary variable
parameter and has a shape that is independent of that parameter. Depending on
an involved parameter, the potential presents either a short-range singular
well (which behaves as inverse square root at the origin and vanishes
exponentially at infinity) or a smooth asymmetric step-barrier (with variable
height and steepness). The general solution of the Schr\"odinger equation for
this potential, which is a member of a general Heun family of potentials, is
written through fundamental solutions each of which presents an irreducible
linear combination of two Gauss ordinary hypergeometric functions.
