the Hermite, Laguerre, and Chebyshev polynomials
/ air-MEET, lah-GAIR, CHEB-ih-shev /
The Legendre polynomials are one member of a larger family: the classical orthogonal polynomials, each arising from its own special second-order equation, each tailored to a different physical setting and a different interval. Three of the most important are the Hermite, Laguerre, and Chebyshev polynomials. They share the Legendre recipe — a series solution that terminates into a polynomial when a parameter is a non-negative integer — but live on different domains and carry different weights.
Take them one at a time. The Hermite polynomials H_n come from y'' - 2x y' + 2n y = 0 and live on the whole real line, orthogonal with the Gaussian weight e^(-x^2); they are the shapes of the quantum harmonic oscillator's energy states, so every textbook picture of a vibrating molecule's wavefunction is built from them (H_0 = 1, H_1 = 2x, H_2 = 4x^2 - 2). The Laguerre polynomials L_n come from x y'' + (1 - x) y' + n y = 0 on the half-line x >= 0, orthogonal with weight e^(-x); they form the radial part of the hydrogen atom's wavefunctions (L_0 = 1, L_1 = 1 - x). The Chebyshev polynomials T_n come from (1 - x^2) y'' - x y' + n^2 y = 0 on [-1, 1], orthogonal with weight 1/sqrt(1 - x^2), and satisfy the striking identity T_n(cos theta) = cos(n theta); they are the heroes of approximation theory and numerical analysis (T_0 = 1, T_1 = x, T_2 = 2x^2 - 1).
The unifying thread is orthogonality with a weight: each family makes the integral of (weight) times P_m times P_n vanish for m not equal n, which lets you expand functions in that family the way Fourier series expands in sines and cosines. The differences — interval and weight — are not arbitrary; each is dictated by the physics or geometry that produced the equation, which is why the right family for a problem is usually the one whose natural domain matches the problem's.
Chebyshev's T_n(cos theta) = cos(n theta) makes its zeros trivially findable: T_4 has zeros where cos(4 theta) = 0, i.e. at x = cos((2k+1) pi/8). Spacing approximations at these 'Chebyshev nodes' nearly eliminates the wild oscillation (Runge's phenomenon) that plagues equally-spaced polynomial interpolation.
Each family shines in its own arena — Hermite in quantum oscillators, Laguerre in the hydrogen atom, Chebyshev in numerical approximation.
These families differ mainly in interval and weight, not in spirit; a common pitfall is to use the wrong weight when computing expansion coefficients — Hermite needs e^(-x^2), Laguerre needs e^(-x), Chebyshev needs 1/sqrt(1-x^2), and getting it wrong destroys orthogonality.