球上本征模的一个新基
A new basis for eigenmodes on the Sphere
摘要：
通常的球谐函数$Y_{\ell m}$形成球面上拉普拉斯算子的本征函数的向量空间${\cal V} ^{\ell}$（维度为$2\ell+1$）的基，具有本征值$λ_{\ell} = -\ell ~(\ell +1)$。在这里，我们表明存在一个不同的基础$Φ^{\ell}_j$的${\cal V} ^{\ell}$，其中$Φ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$，$\ell ^{th}$功率的标量积的当前点与一个特定的零矢量$N_j$。我们明确地给出了这两种基之间的变换性质。在新的基础上计算的简单性允许容易操纵的谐波函数。特别是，我们表示的变换规则的新的基础上，在任何等距的领域。将通常的谐波$Y_{\ell m}$发展成新的基（和反向）允许导出$Y_{\ell m}$的新性质。特别是，这导致了一个新的关系的$Y_{\ell m}$，这是一个有限的版本，众所周知的整表示公式。它还为勒让德多项式和特殊的勒让德函数提供了新的展开公式。
英文摘要：
The usual spherical harmonics $Y_{\ell m}$ form a basis of the vector space ${\cal V} ^{\ell}$ (of dimension $2\ell+1$) of the eigenfunctions of the Laplacian on the sphere, with eigenvalue $λ_{\ell} = -\ell ~(\ell +1)$. Here we show the existence of a different basis $Φ^{\ell}_j$ for ${\cal V} ^{\ell}$, where $Φ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$, the $\ell ^{th}$ power of the scalar product of the current point with a specific null vector $N_j$. We give explicitly the transformation properties between the two bases. The simplicity of calculations in the new basis allows easy manipulations of the harmonic functions. In particular, we express the transformation rules for the new basis, under any isometry of the sphere. The development of the usual harmonics $Y_{\ell m}$ into thee new basis (and back) allows to derive new properties for the $Y_{\ell m}$. In particular, this leads to a new relation for the $Y_{\ell m}$, which is a finite version of the well known integral representation formula. It provides also new development formulae for the Legendre polynomials and for the special Legendre functions.
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