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,
\[=1 + xz + \frac{1}{2}(3x^2-1)z^2 + \frac{1}{2}(5x^3-3x)z^3 + \cdots\]
In this way, Legendre Polynomials are given.
The first few Legendre Polynomials are:
\[\begin{eqnarray}
P_0[x] & = & 1 \\
P_1[x] & = & x \\
P_2[x] & = & \frac{1}{2}(3x^2-1) \\
P_3[x] & = & \frac{1}{2}(5x^3-3x) \\
P_4[x] & = & \frac{1}{8}(35x^4-30x^2+3) \\
P_5[x] & = & \frac{1}{8}(63x^5-70x^3+15x) \\
\end{eqnarray}\]
[1] Legendre, A. M. (1817). Exercices de calcul intégral. tome II. pp.247-248