论文标题
关于Legendre多项式的讲义：其起源和主要特性
Lecture notes on Legendre polynomials: their origin and main properties
论文作者
论文摘要
众所周知，对于球形对称性的物理问题，第二阶部分偏微分方程（PDE）的变量的分离通常会导致Cauchy的径向坐标和Legendre的微分方程的极性角度$θ$。对于表单$ \，n \，（n+1）$，$ n \ ge 0 \，$作为整数的特征值，Legendre的方程式承认某些polyenmials $ p_n（\cosθ）为解决方案，构成了一组连续的正式函数，可用于所有$θ\ in [0，[0，[0，π]。这使我们可以将多项式$ p_n（x）$，其中$ x = \cosθ$作为任何功能的傅立叶 - legendre系列扩展的基础，$ f（x）$连续由$ \，x \ in [-1,1] $连续零件。这些讲义的注释对应于我关于物理学数学方法的结束，当我确实得出了针对球形对称性物理问题的微分方程和解决方案时。对于那些对数字理论感兴趣的人，我在\ emph {非理性的证明}中包括了转移的Legendre多项式的应用，遵循Beukers介绍的方法表明$ζ{（2）} $和$ζ{（3）} $是非理性的数字。
It is well-known that separation of variables in 2nd order partial differential equations (PDEs) for physical problems with spherical symmetry usually leads to Cauchy's differential equation for the radial coordinate and Legendre's differential equation for the polar angle $θ$. For eigenvalues of the form $\,n\,(n+1)$, $n \ge 0\,$ being an integer, Legendre's equation admits certain polynomials $P_n(\cosθ)$ as solutions, which form a complete set of continuous orthogonal functions for all $θ\in [0,π]$. This allows us to take the polynomials $P_n(x)$, where $x = \cosθ$, as a basis for the Fourier-Legendre series expansion of any function $f(x)$ continuous by parts over $\,x \in [-1,1]$. These lecture notes correspond to the end of my course on Mathematical Methods for Physics, when I did derive the differential equations and solutions for physical problems with spherical symmetry. For those interested in Number Theory, I have included an application of shifted Legendre polynomials in \emph{irrationality proofs}, following a method introduced by Beukers to show that $ζ{(2)}$ and $ζ{(3)}$ are irrational numbers.
