In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form These polynomials, usually denoted , , …, are a polynomial sequence which may be defined by the Rodrigues formula, reducing to the closed form of a following section. They are orthogonal polynomials with respect to an inner product The sequence of Laguerre polynomials is a Sheffer sequence, The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.) The first few polynomials These are the first few Laguerre polynomials: Recursive definition, closed form, and generating function One can also define the Laguerre polynomials recursively, defining the first two polynomials as and then using the following recurrence relation for any : Furthermore, In solution of some boundary value problems, the characteristic values can be useful: The closed form is The generating function for them likewise follows, Polynomials of negative index can be expressed using the ones with positive index: Relation to binary functions There is a method to set Laguerre polynomials using functions which is related to binary expansion of : Here with . Also Here is and is a generalisation of . Generalized Laguerre polynomials For arbitrary real α the polynomial solutions of the differential equation