Modified Spherical Harmonics in Four Dimensions

Abstract

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 $S_{+}$, 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 $\mathbb {R}^4 =\left\{ (x,y,t,s) \right\}$ the Laplace equation $\Delta h=0$ will be replaced by the equation $s\Delta u+2\,\frac{\partial u}{\partial s}=0$. Homogeneous polynomial solutions of this equation, if restricted to the half-sphere $S_{+}=\left\{ (x,y,t,s): x^2 + y^2 + t^2 + s^2=1, s > 0 \right\}$ are called modified spherical harmonics. Endowed with a non-Euclidean scalar product on $S_{+}$, these functions behave like the classical spherical harmonics on the full sphere in $\mathbb {R}^4$. 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.