Legendre Polynomial/Examples/P2

Example of Legendre Polynomial

The $2$nd Legendre polynomial is:

$\map {P_2} x = \dfrac 1 2 \paren {3 x^2 - 1}$

Proof

\(\ds \forall n \in \N: \, \)

\(\ds \paren {n + 1} \map {P_{n + 1} } x\)

\(=\)

\(\ds \paren {2 n + 1} x \map {P_n} x - n \map {P_{n - 1} } x\)

Bonnet's Recursion Formula

\(\ds \leadsto \ \ \)

\(\ds 2 \map {P_2} x\)

\(=\)

\(\ds \paren {2 \times 1 + 1} x \map {P_1} x - 1 \map {P_0} x\)

setting $n = 1$

\(\ds \)

\(=\)

\(\ds 3 x \map {P_1} x - \map {P_0} x\)

simplifying

\(\ds \)

\(=\)

\(\ds 3 x \paren x - 1\)

Legendre Polynomial $P_0$ and Legendre Polynomial $P_1$

\(\ds \leadsto \ \ \)

\(\ds \map {P_2} x\)

\(=\)

\(\ds \dfrac 1 2 \paren {3 x^2 - 1}\)

simplifying

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generating function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generating function