  Hahn polynomials of several variables can be defined by using the Jacobi
polynomials on the simplex as a generating function. Starting from this
connection, a number of properties for these two families of orthogonal
polynomials are derived. It is shown that the Hahn polynomials appear as
connecting coefficients between several families of orthogonal polynomials on
the simplex. Closed-form formulas are derived for the reproducing kernels of
the Hahn polynomials and Krawtchouck polynomials. As an application, the
Poisson kernels for the Hahn polynomials and the Krawtchouck polynomials are
shown to be nonnegative.
