  In this paper we discuss the $R$-matrix approach to treat the subthreshold
resonances for the single-level and one channel, and for the single-level and
two channel cases. In particular, the expression relating the ANC with the
observable reduced width, when the subthreshold bound state is the only channel
or coupled with an open channel, which is a resonance, is formulated. Since the
ANC plays a very important role in nuclear astrophysics, these relations
significantly enhance the power of the derived equations. We present the
relationship between the resonance width and the ANC for the general case and
consider two limiting cases: wide and narrow resonances. Different equations
for the astrophysical S-factors in the $R$-matrix approach are presented. After
that we discuss the Trojan Horse Method (THM) formalism. The developed
equations are obtained using the surface-integral formalism and the generalized
$R$-matrix approach for the three-body resonant reactions. It is shown how the
Trojan Horse (TH) double differential cross section can be expressed in terms
of the on-the-energy-shell astrophysical S-factor for the binary sub-reaction.
Finally, we demonstrate how the THM can be used to calculate the astrophysical
S-factor for the neutron generator $^{13}{\rm C}(\alpha,\,n)^{16}{\rm O}$ in
low-mass AGB stars. At astrophysically relevant energies this astrophysical
S-factor is controlled by the threshold level $1/2^{+}, E_{x}= 6356$ keV. Here,
we reanalyzed recent TH data taking into account more accurately the three-body
effects and using both assumptions that the threshold level is a subthreshold
bound state or it is a resonance state.
