多调和方程幂律基本解的Fourier、Gegenbauer和Jacobi展开式及多球加法定理
Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems
摘要：
我们开发复杂的Jacobi，Gegenbauer和Chebyshev多项式展开的核与幂律基本解的多重调和方程的d维欧氏空间。从这些系列表示，我们得到傅立叶展开在某些旋转不变的坐标系和Gegenbauer多项式展开Vilenkin的多球坐标系。我们比较这两个扩展生成的方位角傅立叶系数的加法定理。
英文摘要：
We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients.
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