  We introduce and develop the notion of spherical polyharmonics, which are a
natural generalisation of spherical harmonics. In particular we study the
theory of zonal polyharmonics, which allows us, analogously to zonal harmonics,
to construct Poisson kernels for polyharmonic functions on the union of rotated
balls. We find the representation of Poisson kernels and zonal polyharmonics in
terms of the Gegenbauer polynomials. We show the connection between the
classical Poisson kernel for harmonic functions on the ball, Poisson kernels
for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua
kernel for holomorphic functions on the Lie ball.
