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Legendre's equation (1-x²)y'' - 2xy' + n(n+1)y = 0 has polynomial solutions P_n(x) for non-negative integers n. These arise in spherically symmetric problems and are orthogonal on [-1,1] with respect to the standard inner product. Legendre polynomials have generating functions, recurrence relations, and explicit formulas, forming the basis for Legendre expansions.
The Legendre equation arises naturally when you solve Laplace's equation ∇²φ = 0 in spherical coordinates and separate variables. The angular part of the solution leads to (1−x²)y'' − 2xy' + n(n+1)y = 0, where x = cos(θ) is the polar angle variable. This is a second-order linear ODE with non-constant coefficients and ordinary points everywhere on (−1, 1), so the Frobenius method you learned tells you to seek a power series solution y = Σ aₖxᵏ around x = 0.
Substituting into the equation and collecting powers of x gives a recurrence relation: aₖ₊₂ = −[n(n+1) − k(k+1)] / [(k+2)(k+1)] · aₖ. The series is determined by the choice of a₀ and a₁ (giving two independent solutions). Here is the key observation: if n is a non-negative integer, the recurrence terminates — the coefficient aₙ₊₂ = 0, and all subsequent coefficients vanish. The series becomes a polynomial of degree n. These terminating solutions are the Legendre polynomials P_n(x), normalized so P_n(1) = 1. The first few are: P₀(x) = 1, P₁(x) = x, P₂(x) = (3x²−1)/2, P₃(x) = (5x³−3x)/2.
The orthogonality of Legendre polynomials is what makes them useful: ∫₋₁¹ Pₘ(x)Pₙ(x) dx = 0 whenever m ≠ n. This mirrors how sine and cosine functions are orthogonal on [−π, π] — the key property that makes Fourier series work. Because Legendre polynomials are orthogonal, any reasonable function f on [−1, 1] can be expanded as f(x) = Σ cₙ Pₙ(x), where the coefficients are extracted by the inner product: cₙ = [(2n+1)/2] ∫₋₁¹ f(x) Pₙ(x) dx. This is a Legendre expansion — the spherical analogue of a Fourier series.
A practical tool is the recurrence relation between consecutive polynomials: (n+1)Pₙ₊₁(x) = (2n+1)x Pₙ(x) − n Pₙ₋₁(x). This lets you compute any Pₙ efficiently without re-deriving the series. Legendre polynomials appear throughout mathematical physics — gravitational and electrostatic potentials in spherical geometry, quantum mechanics of the hydrogen atom, and heat conduction on spheres — wherever physical symmetry makes spherical coordinates natural.