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Chemistry LibreTexts

M2: Legendre Polynomials

Each Legendre polynomial \(P_n(x)\,\) is an \(n\)=th-degree polynomial. It may be expressed using Rodrigues' formula:

\[P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right] \]

That these polynomials satisfy the Legendre differential equation follows by differentiating (n+1) times both sides of the identity

\[(x^2-1)\frac{d}{dx}(x^2-1)^n = 2nx(x^2-1)^n\]

The first few Legendre polynomials are:

\(n\) \(P_n(x)\,\)
0 \(1\)
1 \(x\)
2 \(\begin{matrix}\frac12\end{matrix} (3x^2-1) \,\)
3 \(\begin{matrix}\frac12\end{matrix} (5x^3-3x) \,\)
4 \(\begin{matrix}\frac18\end{matrix} (35x^4-30x^2+3)\,\)
5 \(\begin{matrix}\frac18\end{matrix} (63x^5-70x^3+15x)\,\)
6 \(\begin{matrix}\frac1{16}\end{matrix} (231x^6-315x^4+105x^2-5)\,\)
7 \(\begin{matrix}\frac1{16}\end{matrix} (429x^7-693x^5+315x^3-35x)\,\)
8 \(\begin{matrix}\frac1{128}\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\,\)
9 \(\begin{matrix}\frac1{128}\end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x)\,\)
10 \(\begin{matrix}\frac1{256}\end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)\,\)