Hermite polynomials
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The Legendre polynomials live on a finite interval; but many physical problems live on the whole real line and care most about behaviour near a peak that fades like a Gaussian bell. The Hermite polynomials are the family of polynomials orthogonal on the entire line with respect to the Gaussian weight e^{-x^2} — the natural basis whenever a bell-shaped envelope is in play.
In the physicists' convention they are H_0 = 1, H_1 = 2x, H_2 = 4 x^2 minus 2, H_3 = 8 x^3 minus 12 x, with H_n of degree n. Their orthogonality reads integral from minus infinity to infinity of H_m(x) H_n(x) e^{-x^2} dx = 0 for m not equal to n — the weight e^{-x^2} is what tames the growing polynomials at infinity. They follow Rodrigues' formula H_n(x) = (-1)^n e^{x^2} times the n-th derivative of e^{-x^2}, the recurrence H_{n+1} = 2 x H_n minus 2 n H_{n-1}, and a generating function. A separate probabilists' convention uses the weight e^{-x^2/2} and gives slightly different polynomials He_n.
Hermite polynomials are the algebraic skeleton of the quantum harmonic oscillator: the energy eigenstates are H_n(x) times a Gaussian, so every vibrating molecule, every photon mode, and the formalism of quantum field theory carries them. They also generate the Hermite-Gaussian transverse modes of a laser beam and underlie Gauss-Hermite numerical integration. Catalogued here as functions; the oscillator differential equation they satisfy belongs to the series-solutions topic.
The ground state of the quantum harmonic oscillator uses H_0 = 1, the first excited state H_1 = 2x; the single node of H_1 (a zero at x = 0) is the one point where the first excited wavefunction vanishes.
The number of zeros of H_n equals n, which becomes the number of nodes of the n-th oscillator state.
Two conventions coexist: physicists' H_n (weight e^{-x^2}) and probabilists' He_n (weight e^{-x^2/2}). They differ by scaling, so always check which one a formula assumes.