Laguerre Polynomials

Laguerre polynomials are a sequence of polynomials named after the French mathematician Edmond Laguerre, arising as the polynomial solutions of the differential equation x y double-prime plus (1 minus x) y prime plus n y equals zero when n is a non-negative integer. A more general version of the equation, replacing (1 minus x) with (alpha plus 1 minus x), produces the generalized or associated Laguerre polynomials. The polynomials form an orthogonal sequence with respect to a weighting function of e to the minus x on the interval from zero to infinity, and can also be constructed through a Rodrigues formula. They are used in Gauss-Laguerre quadrature for numerically evaluating integrals of the form the integral from zero to infinity of f(x) e to the minus x dx, and they appear in the solutions of quantum mechanical problems such as the hydrogen atom.