An elementary solution of the biharmonic equation is defined. By using the properties of the Gegenbauer polynomials, series expansions of this elementary solution and an associated function with respect to a complete system of homogeneous harmonic polynomials orthogonal on a unit sphere are obtained. Then the Green function of the Dirichlet problem for the biharmonic equation in a unit ball is constructed in the case when the space dimension n is larger than 2. For $$n > 4$$ , a series expansion of the Green function with respect to a complete system of homogeneous harmonic polynomials orthogonal on a unit sphere is obtained. This expansion is used to calculate the integral, over a unit ball, of a homogeneous harmonic polynomial multiplied by a positive power of the norm of the independent variable with a kernel being the Green function. The Green function is found in the case $$n = 2$$ .