Matches in DBpedia 2014 for { ?s ?p In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 – 1886),are solutions of Laguerre's equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. The associated Laguerre polynomials (alternatively, but rarely, named Sonin polynomials, after their inventor N. Y. Sonin) are solutions of The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form These polynomials, usually denoted L0, L1, ..., 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 n! Ln is a Sheffer sequence, d⁄dx Ln = (d⁄dx−1) Ln−1.The Rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables.The Laguerre polynomials arise in quantum mechanics, in the radial part of the solutionof 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 3D isotropic harmonic oscillator.Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of n! than the definition used here. (Likewise, some physicist may use somewhat different definitions of the so-called associated Laguerre polynomials.). }
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- Laguerre_polynomials abstract "In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 – 1886),are solutions of Laguerre's equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. The associated Laguerre polynomials (alternatively, but rarely, named Sonin polynomials, after their inventor N. Y. Sonin) are solutions of The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form These polynomials, usually denoted L0, L1, ..., 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 n! Ln is a Sheffer sequence, d⁄dx Ln = (d⁄dx−1) Ln−1.The Rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables.The Laguerre polynomials arise in quantum mechanics, in the radial part of the solutionof 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 3D isotropic harmonic oscillator.Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of n! than the definition used here. (Likewise, some physicist may use somewhat different definitions of the so-called associated Laguerre polynomials.)".