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Laguerre Polynomial — Definition, Formula & Examples

Laguerre polynomials are a sequence of orthogonal polynomials that arise as solutions to the Laguerre differential equation xy′′+(1−x)y′+ny=0xy'' + (1-x)y' + ny = 0, where nn is a non-negative integer.

The nnth Laguerre polynomial Ln(x)L_n(x) is defined by the closed-form expression Ln(x)=∑k=0n(nk)(−x)kk!L_n(x) = \sum_{k=0}^{n} \binom{n}{k} \frac{(-x)^k}{k!}, or equivalently by the Rodrigues formula Ln(x)=exn!dndxn(xne−x)L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n}(x^n e^{-x}). These polynomials are orthogonal on [0,∞)[0, \infty) with respect to the weight function e−xe^{-x}.

Key Formula

Ln(x)=∑k=0n(nk)(−x)kk!L_n(x) = \sum_{k=0}^{n} \binom{n}{k} \frac{(-x)^k}{k!}
Where:
  • nn = Non-negative integer giving the degree of the polynomial
  • xx = The variable, typically defined on the interval [0, ∞)
  • (nk)\binom{n}{k} = Binomial coefficient 'n choose k'

How It Works

Each Laguerre polynomial Ln(x)L_n(x) has degree nn and leading coefficient (−1)nn!\frac{(-1)^n}{n!}. The first few are L0(x)=1L_0(x) = 1, L1(x)=1−xL_1(x) = 1 - x, and L2(x)=1−2x+12x2L_2(x) = 1 - 2x + \tfrac{1}{2}x^2. They satisfy a three-term recurrence relation: (n+1)Ln+1(x)=(2n+1−x)Ln(x)−nLn−1(x)(n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x), which is the most efficient way to compute higher-degree Laguerre polynomials. Their orthogonality means ∫0∞Lm(x)Ln(x)e−x dx=0\int_0^{\infty} L_m(x) L_n(x) e^{-x}\,dx = 0 whenever m≠nm \neq n.

Worked Example

Problem: Compute L3(x)L_3(x) using the summation formula.
Step 1: Write out the sum for n=3n = 3 with k=0,1,2,3k = 0, 1, 2, 3.
L3(x)=(30)(−x)00!+(31)(−x)11!+(32)(−x)22!+(33)(−x)33!L_3(x) = \binom{3}{0}\frac{(-x)^0}{0!} + \binom{3}{1}\frac{(-x)^1}{1!} + \binom{3}{2}\frac{(-x)^2}{2!} + \binom{3}{3}\frac{(-x)^3}{3!}
Step 2: Evaluate each binomial coefficient and simplify.
L3(x)=1⋅1+3⋅−x1+3⋅x22+1⋅−x36L_3(x) = 1 \cdot 1 + 3 \cdot \frac{-x}{1} + 3 \cdot \frac{x^2}{2} + 1 \cdot \frac{-x^3}{6}
Step 3: Combine the terms.
L3(x)=1−3x+32x2−16x3L_3(x) = 1 - 3x + \frac{3}{2}x^2 - \frac{1}{6}x^3
Answer: L3(x)=1−3x+32x2−16x3L_3(x) = 1 - 3x + \frac{3}{2}x^2 - \frac{1}{6}x^3

Why It Matters

Laguerre polynomials appear in the radial part of the hydrogen atom wave function in quantum mechanics, making them essential in physics courses. They also serve as basis functions in Gauss-Laguerre quadrature, a numerical method for evaluating integrals over [0,∞)[0, \infty) with exponential-type integrands.

Common Mistakes

Mistake: Forgetting the 1k!\frac{1}{k!} factor and treating the formula as a standard binomial expansion.
Correction: Each term has both a binomial coefficient (nk)\binom{n}{k} and a factorial denominator k!k!. These do not cancel — double-check by verifying L1(x)=1−xL_1(x) = 1 - x, not 1−x1 - x times some extra factor.