A modification of the classical theory of spherical harmonics in four dimensions is presented. The space R4={(x,y,t,s)}\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^4 = \{(x,y,t,s)\}$$\end{document} is replaced by the upper half space R+4=(x,y,t,s),s>0\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}_{+}^{4}=\left\{ (x,y,t,s), s > 0 \right\} $$\end{document}, and the unit sphere S in R4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^4$$\end{document} by the unit half sphere S+=(x,y,t,s):x2+y2+t2+s2=1,s>0\documentclass[12pt]{minimal}
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\begin{document}$$S_{+}=\left\{ (x,y,t,s): x^2 +y^2+ t^2+ s^2 =1, s > 0 \right\} $$\end{document}. Instead of the Laplace equation Δh=0\documentclass[12pt]{minimal}
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\begin{document}$$\Delta h = 0$$\end{document} we shall consider the Weinstein equation sΔu+k∂u∂s=0\documentclass[12pt]{minimal}
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\begin{document}$$s\Delta u + k \frac{\partial u }{\partial s}= 0$$\end{document}, for k∈N\documentclass[12pt]{minimal}
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\begin{document}$$k \in \mathbb {N}$$\end{document}. The Euclidean scalar product for functions on S will be replaced by a non-Euclidean one for functions on S+\documentclass[12pt]{minimal}
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\begin{document}$$S_{+}$$\end{document}. 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.
