A new class of three-variable orthogonal polynomials and their recurrences relations
The author studies a new class of three variable orthogonal polynomials, defined as eigenfunctions of a second order partial differential operator. 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 2D 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 are now determined to be five and seven, with respect to two conjugate variables \(z,\bar{z}\) and a real variable \(r\), respectively. Three examples are discussed in detail, which can be regarded as the analogues of the Chebyshev polynomials of first and second kinds, and Legendre polynomials.
scientific article; zbMATH DE number 5323379