  For the well-known exponential complexity it is a giant challenge to
calculate the correlation function for general many-body wave function. We
investigate the ground state $n$th-order correlation functions of the
Tonks-Girardeau (TG) gases. Basing on the wavefunction of free fermions and
Bose-Fermi mapping method we obtain the exact ground state wavefunction of TG
gases. Utilizing the properties of Vandermonde determinant and Toeplitz matrix,
the $n$th-order correlation function is formulated as $(N-n)$-order Toeplitz
determinant, whose element is the integral dependent on 2$(N-n)$ sign functions
and can be computed analytically. By reducing the integral on domain $[0,2\pi]$
into the summation of the integral on several independent domains, we obtain
the explicit form of the Toeplitz matrix element ultimately. As the
applications we deduce the concise formula of the reduced two-body density
matrix and discuss its properties. The corresponding natural orbitals and their
occupation distribution are plotted. Furthermore, we give a concise formula of
the reduced three-body density matrix and discuss its properties. It is shown
that in the successive second measurements, atoms appear in the regions where
atoms populate with the maximum probability in the first measurement.
