The classical theory of spherical harmonics on the unit sphere S is well-known. In an earlier paper, entitled “Modified Spherical Harmonics”, we dealt with a modification of this theory, adapted to the half-sphere S+\documentclass[12pt]{minimal}
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\begin{document}$$S_{+}$$\end{document}, in case of three dimensions. In the present paper we extend these results to the four-dimensional case. Although the results look quite similar, their proofs are not. In R4=(x,y,t,s)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^4 =\left\{ (x,y,t,s) \right\} $$\end{document} the Laplace equation Δh=0\documentclass[12pt]{minimal}
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\begin{document}$$\Delta h=0$$\end{document} will be replaced by the equation sΔu+2∂u∂s=0\documentclass[12pt]{minimal}
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\begin{document}$$s\Delta u+2\,\frac{\partial u}{\partial s}=0$$\end{document}. Homogeneous polynomial solutions of this equation, if restricted to the 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} are called modified spherical harmonics. Endowed with a non-Euclidean scalar product on S+\documentclass[12pt]{minimal}
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\begin{document}$$S_{+}$$\end{document}, these functions behave like the classical spherical harmonics on the full sphere in R4\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^4$$\end{document}. We shall give an explicit expression for the corresponding zonal harmonics and dwell on their connection with the Poisson-type kernel, adapted to the above differential equation. Finally we shall give an explicit orthonormal system of modified spherical harmonics.
