Legendre Polynomials
Legendre Polynomials are the coefficients of the power series of the following generator function: [1] \[\frac{1}{\sqrt{1-2xz+z^2}} = \sum\limits_{n=0}^{\infty}P_l[x]z^n\] where \(P_l[x]\) is Legendre Polynomials at \(l\) th order. Taylor series is: \[\frac{1}{\sqrt{1-(2xz-z^2)}} = 1 + \frac{1}{2}(2xz-z^2) + \frac{1\cdot 3}{2\cdot 4}(2xz-z^2)^2 + \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}(2xz-z^2)^3 + \cdots\] after rearranging, \[=1 + xz + \frac{1}{2}(3x^2-1)z^2 + […]