A modification of the classical theory of spherical harmonics is presented. The unit sphere S in R3={(x,y,t)}\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{R}^3 = \{(x,y,t)\}}$$\end{document} is replaced by the half-sphere S+ in the upper half space, the Euclidean scalar product on S by a non-Euclidean one on S+, and the Laplace equation Δh=0\documentclass[12pt]{minimal}
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\begin{document}$${\Delta h = 0}$$\end{document} by the equation tΔv+∂v∂t=0\documentclass[12pt]{minimal}
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\begin{document}$${t\Delta v + \frac{\partial v }{\partial t}= 0}$$\end{document}. It will be shown that most results from the theory of spherical harmonics in R3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{R}^3}$$\end{document} stay valid in this modified setting.
