A new class of three-variable orthogonal polynomials and their recurrences relations

A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, (z) over bar and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.