论文标题
关于超球相关的Legendre功能：将球形谐波扩展到$ n $ dimensions
On hyperspherical associated Legendre functions: the extension of spherical harmonics to $N$ dimensions
论文作者
论文摘要
对于$ n $尺寸的超透明坐标的解决方案是针对一类数学物理学的一般偏微分方程，包括拉普拉斯，波浪，波浪，热和赫尔姆霍尔茨，schrödinger，klein-gordon和电报方程及其组合。起点是由超球坐标的比例因子指定的拉普拉斯运算符。数学物理学的一般方程是通过将变量的分离来求解的：（i）按时按通常的指数函数进行时间； （ii）通过通常的正弦功能在经度上； （iii）在半径上通过贝塞尔函数，通常与圆柱形或球形贝塞尔函数不同； （iv）通过关联的legendre函数在一个纬度上； （v）在其余纬度上通过扩展，即超球相关的legendre函数。原始相关的Legendre函数是高斯高几幅函数的特殊情况，并且超球形相关的Legendre函数也是高斯高几何函数的更一般的情况，因此无需考虑扩展高斯高斯高点功能。
The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schrödinger, Klein-Gordon and telegraph equations and their combinations. The starting point is the Laplacian operator specified by the scale factors of hyperspherical coordinates. The general equation of mathematical physics is solved by separation of variables leading to the dependencies: (i) on time by the usual exponential function; (ii) on longitude by the usual sinusoidal function; (iii) on radius by Bessel functions of order generally distinct from cylindrical or spherical Bessel functions; (iv) on one latitude by associated Legendre functions; (v) on the remaining latitudes by an extension, namely the hyperspherical associated Legendre functions. The original associated Legendre functions are a particular case of the Gaussian hypergeometric functions, and the hyperspherical associated Legendre functions are also a more general particular case of the Gaussian hypergeometric functions so that it is not necessary to consider extended Gaussian hypergeometric functions.
