Vol. 92, No. 1, 1981

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A difference equation and Hahn polynomials in two variables

Charles F. Dunkl

Vol. 92 (1981), No. 1, 57–71

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.

Mathematical Subject Classification 2000
Primary: 33A65, 33A65
Secondary: 39A10
Received: 4 February 1979
Published: 1 January 1981
Charles F. Dunkl