  A lattice Boltzmann scheme associated with flexible Prandtl number and
specific heat ratio is proposed, which is based on the polyatomic ellipsoidal
statistics model(ES-BGK). The Prandtl number can be modified by a parameter of
the Gaussian distribution and the specific heat ratio can be modified by
additional free degrees. For the sake of constructing the scheme proposed, the
Gaussian distribution is expanded on the Hermite polynomials and the general
term formula for the Hermite coefficients of the Gaussian distribution is
deduced. Benchmarks are carried out to verify the scheme proposed. The
numerical results are in good agreement with the the analytical solutions.
