Planar Harmonic Polynomials of Type B

The hyperoctahedral group acting on R is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators {Ti : 1 ≤ i ≤ N}. These operators are analogous to partial derivative operators. This paper finds all the polynomials h on R which are harmonic, ∆Bh = 0 and annihilated by Ti for i > 2, where the Laplacian ∆B = ∑ N i=1 T 2 i . They are given explicitly in terms of a novel basis of polynomials, defined by generating functions. The harmonic polynomials can be used to find wave functions for the quantum many-body spin Calogero model.