Poisson and Cauchy kernels for orthogonal polynomials with dihedral symmetry
For each dihedral subgroup of the orthogonal group on \({\mathbb{R}}^ 2\cong {\mathbb{C}}\) there are a family of measures \(h(e^{i\theta})^ 2\) on the unit circle and associated commuting first-order differential-difference operators \(\bar T_ h\) and \(T_ h\) which act much like \(\partial /\partial \bar z\) and \(\partial /\partial z\). There is an orthogonal basis for \(L^ 2(h(e^{i\theta})^ 2d\theta)\) expressed in terms of Gegenbauer and Jacobi polynomials. These are boundary values of so-called h-harmonic functions on the disk, that is, functions f such that \(T_ h\bar T_ hf(z)=0\), \(| z| <1.\) There is a maximum principle for h-harmonic functions, which implies the positivity of the Poisson kernel, namely that kernel that reproduces such functions from their boundary values. The Cauchy kernel will be defined to be the kernel that reproduces functions in the kernel of \(\bar T_ h\) on the disk, and annihilates the orthogonal complement of such functions in \(L^ 2(h(e^{i\theta})^ 2d\theta).\) The main results of this paper are the explicit determinations in terms of hypergeometric and Appell functions of the Poisson and Cauchy kernels for all the possible measures, namely \((\sin^ 2k\theta)^{\alpha}(\cos^ 2k\theta)^{\beta}d\theta\) with \(\alpha >0\) and \(\beta >0\) when k is even, \(\beta\geq 0\) when k is odd, \(k=1,2,3,..\)..
scientific article; zbMATH DE number 4127617