Laguerre polynomials
/ lah-GAIR /
Some problems live not on a finite interval, nor on the whole line, but on the half-line from 0 to infinity, with a decaying exponential setting the scale — think of the radial direction outward from an atomic nucleus. The Laguerre polynomials are the family of polynomials orthogonal on the positive half-line with respect to the weight e^{-x}, the natural basis for that geometry.
They are L_0 = 1, L_1 = 1 minus x, L_2 = (x^2 minus 4 x plus 2)/2, and so on. Orthogonality reads integral from 0 to infinity of L_m(x) L_n(x) e^{-x} dx = 0 for m not equal to n; the weight e^{-x} both supplies the decay that makes the integral converge and encodes the half-line geometry. They obey Rodrigues' formula L_n(x) = (e^x over n!) times the n-th derivative of (x^n e^{-x}), a three-term recurrence, and a generating function. A generalised family L_n^{alpha}, orthogonal with the weight x^{alpha} e^{-x}, appears when an extra power of x is present.
The generalised Laguerre polynomials are the radial part of the hydrogen atom wavefunction — the precise polynomial in the radius that, multiplied by an exponential, gives the s, p, d orbital shapes. They also drive Gauss-Laguerre numerical integration for integrals over the positive axis with an exponential factor, and appear in vibrational spectra and quantum optics. Listed here among the orthogonal-polynomial families; their generating equation belongs to the series-solutions topic.
Normalisation check: integral from 0 to infinity of L_0(x)^2 e^{-x} dx = integral from 0 to infinity of e^{-x} dx = 1, using the simplest Laguerre polynomial L_0 = 1 and the gamma value Gamma(1) = 1.
The exponential weight e^{-x} is exactly what makes these half-line integrals finite and ties them to the gamma function.
The plain L_n and the generalised L_n^{alpha} are different families; the hydrogen atom needs the generalised version, so do not substitute the simpler one.