Contributions to Modified Spherical Harmonics in Four Dimensions

A modification of the classical theory of spherical harmonics in four dimensions is presented. The space R4={(x,y,t,s)} is replaced by the upper half space R+4</mml:msubsup>=<mml:mfenced close="}" open="{">(x,y,t,s),s>0</mml:mfenced>, and the unit sphere S in R4 by the unit half sphere S+=<mml:mfenced close="}" open="{">(x,y,t,s):x2+y2+t2+s2=1,s>0</mml:mfenced>. Instead of the Laplace equation Delta h=0 we shall consider the Weinstein equation s Delta u+k<mml:mfrac>partial derivative u partial derivative s</mml:mfrac>=0, for k is an element of N. The Euclidean scalar product for functions on S will be replaced by a non-Euclidean one for functions on <mml:msub>S+. It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In addition we shall deduct-with respect to this non-Euclidean scalar product-an orthonormal system of homogeneous polynomials, which satisfies the above Weinstein equation.