论文标题
Gegenbauer的扩展和添加定理，用于均值均值欧几里得多谐调方程的二项式和对数基本解决方案
Gegenbauer expansions and addition theorems for a binomial and logarithmic fundamental solution of the even-dimensional Euclidean polyharmonic equation
论文作者
论文摘要
在均匀的欧几里得空间上，用于拉普拉斯操作员的整数幂大于或等于尺寸的一半，多谐式方程的基本解决方案具有二项式和对数行为。这些基本解决方案的gegenbauer多项式扩展是通过应用于多谐式方程的幂律基本溶液的gegenbauer扩展的限制获得的。通过参数分化来实现此限制。通过将这些结果与这些二项式和对数基本溶液的先前衍生的方位角傅立叶系列膨胀相结合，我们能够获得方位角傅立叶系数的添加定理。这些对数和二项式添加定理在Vilenkin多球测量层坐标系统中表达，以及在任意甚至尺寸的球体上的广义HOPF坐标。
On even-dimensional Euclidean space for integer powers of the Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on spheres in arbitrary even dimensions.
