Recipe

Compute a Legendre polynomial by Rodrigues' formula

Use when: you need $P_n(x)$ itself, or its value at one point, for a specific nonnegative integer $n$

Goal: the polynomial $P_n(x)$ , or its value at the point you need, matching the standard normalization

Taught in Legendre's equation and Legendre polynomials

Quick recipe

Decide whether to derive $P_n$ or just evaluate a known one
Form $(x^2-1)^n$ and differentiate it $n$ times
Divide by $2^nn!$ to get $P_n(x)$
Substitute the target $x$ -value

1
Check whether $n$ is small enough that its closed form is already known or given. Or does it genuinely have to be derived from scratch?
- no closed form is known or given — derive it with Rodrigues' formula, continuing to Step 2.
- the closed form is already known (or handed to you) — skip straight to Step 4 and substitute the target $x$ -value.
Why: Rodrigues' formula always works. But re-differentiating an already-tabulated low-degree polynomial, just to evaluate it at one point, is wasted effort.
2
Expand $(x^2-1)^n$ first. Then differentiate that expanded polynomial $n$ times in a row. Keep track of each intermediate derivative.

Example

For $n=2$ : $(x^2-1)^2=x^4-2x^2+1$ ; first derivative $4x^3-4x$ ; second derivative $12x^2-4$ .

Why: Rodrigues' formula packages the entire derivation of $P_n$ into one repeated differentiation. No series or boundary-value argument is needed.
3
Scale the $n$ -th derivative by $\dfrac1{2^nn!}$ . This scaling is what fixes the standard convention $P_n(1)=1$ .

Example

Dividing $12x^2-4$ by $2^2\cdot2!=8$ gives $P_2(x)=\dfrac{12x^2-4}8=\dfrac{3x^2-1}2$ .

Why: without this specific constant, the derivative alone still solves Legendre's equation. But it would not match the polynomial everyone else calls $P_n$ .
4
Once the polynomial is in hand -- whether derived or already known -- plug in the specific number. Then simplify.

Example

Using $P_3(x)=\dfrac{5x^3-3x}2$ : $P_3(0.5)=\dfrac{5(0.125)-3(0.5)}2=-0.4375$ .

Why: evaluating at a point is ordinary substitution, once the closed-form polynomial is available. Nothing about Legendre's equation is involved at this stage.

Common mistakes

Forgetting the $\tfrac1{2^nn!}$ scale factor and reporting the bare $n$ -th derivative as $P_n(x)$ .
Checking boundedness at only $x=1$ when confirming a solution is the right one. Legendre's equation is singular at BOTH $x=\pm1$ .
Assuming $P_n$ 's parity without checking it: even $n$ gives an even polynomial, odd $n$ gives an odd one. Always check it against the actual derivative.
Re-deriving a low-degree $P_n$ from scratch every time. Recall its closed form instead, when only a single evaluation is needed.