The space of polynomials in N
variables spanned by square-free monomials of degree r and annihilated by
∑i=1N∂∕∂xi furnishes an irreducible representation of SN, the symmetric group on
N objects. The elements of this space, which are invariant under permutations
leaving the sets {x1,⋯,xa} and {xa+1,⋯,xa+b}(a + b < N) fixed, correspond to
solutions of a linear difference equation in two variables. By using ideas of
representation theory, orthogonal bases for the space of solutions can be obtained.
They are certain families of Hahn polynomials in two variables. When these
polynomials are restricted to appropriate subsets of RN, general Hahn polynomials in
two variables (defined by Karlin and McGregor for the study of populations with
various types) are obtained. Further the group theory shows there are three
orthogonal bases for the space of solutions of the difference equation, and the
connection coefficients between different bases turn out to be balanced 4F3-sums,
related to Racah’s 6 − j symbols and Wilson’s four-parameter orthogonal
polynomials.
