  An explicit formula for a new type of beams, which in this work are called
the "special" hyperbolic Bessel-Gaussian (SHBG) beams, has been derived, using
the method of the Hankel transform formulated in our previous work. The
fundamental properties of these beams are analyzed. The parameters that define
the beam shape have been identified and related to those of the fundamental
Gaussian beam. The analytical expressions for the SHBG beams include an
additional parameter $\gamma$, which allows the beam's shape to be modified to
some extent. In the plane perpendicular to the propagation direction, these
beams exhibit the annular nature. Interestingly, initially (i.e. near the
beam's spot) a single ring splits into a number of rings as one is moving along
the beam. This is especially apparent for $\gamma$ close to unity, as this
effect then appears for values of $z$ relatively small compared to the Rayleigh
length i.e., where the energy concentration in the beam is still high. The
phase of the wave, whose behavior is in certain aspects typical of modes having
the vortex character, is also studied in this paper.
