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A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others. Introduction For the investigation of harmonic polynomials a scalar product for homogeneous polynomials of degree m in the form 〈Pm(x), Qm(x)〉 = Pm(D)Qm(x) was introduced in [1]—one of the basic works on harmonic analysis— where the operator Pm(D) is obtained from the polynomial Pm(x) by replacing each variable xi on the differential operator ∂/∂xi. If we denote a set of all polynomials over C by P , then that scalar product can be extended on P in the following way: 〈P (x), Q(x)〉 = P (D)Q(x)|x=0. This idea proved to be very successful for the investigation of polynomial solutions to systems of PDE with constant coefficients (see for instance [2], [3]). In the present work we shall consider a full set of harmonic polynomials which are furthermore orthogonal both in L2(∂Sn) (Sn is a unit ball in R) and in P , and we shall give one interesting property of theirs (Corollary 2). 1. System of harmonic polynomials G(ν)(x) Let k, s ∈ N (N ≡ N ∪ {0}) and n ∈ N but n > 1. Consider the polynomials

On one set of orthogonal harmonic polynomials