Moments of measure orthogonalizing the 2-dimensional Chebyshev polynomials

We calculate the moments mk,0 of the measure orthogonalizing the 2-dimensional Chebyshev polynomials introduced by Koornwinder. In [K] Koornwinder introduced a two-dimensional analogue of the classical Chebyshev polynomials of the second kind. They are defined by the following recurrence relations: P −1,l(z, z̄) = 0, Pk,−1(z, z̄) = 0 P0,0(z, z̄) = 1, P1,0(z, z̄) = z, P0,1(z, z̄) = z̄, z Pk,l(z, z̄) = Pk+1,l(z, z̄) + Pk−1,l+1(z, z̄) + Pk,l−1(z, z̄),(1) z ̄ Pk,l(z, z̄) = Pk,l+1(z, z̄) + Pk+1,l−1(z, z̄) + Pk−1,l(z, z̄)(2) The total degree of Pk,l(z, z̄) is thus k + l. For general properties of multidi-mensional orthogonal polynomials see for instance [DX]. Those polynomials form a system orthonormal with respect to the weight function µ(z, z̄) µ(z, z̄) =