Chebyshev, Legendre, Hermite and Other Orthonormal Polynomials in D Dimensions

We propose a set of polynomials orthonormal under a general weight which are symmetrical tensors in D-dimensional Euclidean space. The D-dimensional Hermite polynomials are shown to be a particular case of the present ones for the case of a Gaussian weight. We explicitly determine the parameters of the first five polynomials (N from 0 to 4) and conjecture that our procedure can be generalized to N-th order because of the remarkable match found between the orthonormality conditions and the symmetrical tensors in the D-dimensional Euclidean space. In this way we obtain generalizations of the Legendre and of the Chebyshev polynomials in D dimensions that reduce to the respective well-known orthonormal polynomials in D = 1 dimensions. We also obtain new D-dimensional polynomials orthonormal under weights of interest to physics, such as the Fermi-Dirac, Bose-Einstein, graphene equilibrium distribution functions and the Yukawa potential.