Laguerre polynomials
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| Laguerre polynomials | |
|---|---|
| Name | Laguerre polynomials |
| Domain | Mathematics |
| Field | Orthogonal polynomials |
| Introduced | 19th century |
| Notable contributors | Edmond Laguerre, Carl Friedrich Gauss, Joseph Fourier |
Laguerre polynomials are a sequence of orthogonal polynomials that arise in solutions of certain second-order linear differential equations and spectral problems. They play central roles in mathematical physics, approximation theory, and computational methods developed in the 19th and 20th centuries alongside contributions from Edmond Laguerre, Carl Friedrich Gauss, and Joseph Fourier. Their properties connect to classical subjects studied by Bernhard Riemann, Augustin-Louis Cauchy, and Lord Kelvin, and they appear in applications pioneered by Paul Dirac, Erwin Schrödinger, and Subrahmanyan Chandrasekhar.
Definition and Basic Properties
A Laguerre polynomial of nonnegative integer order n is commonly defined as a polynomial of degree n with coefficients determined by combinatorial expressions studied by Pierre-Simon Laplace, Adrien-Marie Legendre, and Sophie Germain; these coefficients are related to binomial coefficients used by Blaise Pascal and Édouard Lucas. The sequence satisfies normalization conventions found in works associated with Henri Poincaré, David Hilbert, and Richard Courant, and its leading coefficient equals (−1)^n/n! consistent with sign conventions used by Augustin-Louis Cauchy, Émile Picard, and Niels Henrik Abel. Their zeros are real, simple, and interlace properties analogous to results proved by Gauss, G. H. Hardy, and Srinivasa Ramanujan.
Orthogonality and Weight Function
Laguerre polynomials are orthogonal on the half-line with respect to an exponential weight investigated in contexts by Joseph Fourier, Bernhard Riemann, and Émile Borel. The orthogonality relations tie to integral transforms developed by Lord Kelvin, Hermann Weyl, and John von Neumann and are essential in spectral theory linked to David Hilbert, John von Neumann, and Laurent Schwartz. Orthogonality is used in expansions analogous to Fourier series applied by Jean-Baptiste Joseph Fourier, Mary Cartwright, and Norbert Wiener in problems studied by Alan Turing and John Nash.
Rodrigues' Formula and Generating Functions
Rodrigues' formula expresses each polynomial via differentiation operations reminiscent of techniques by Siméon Denis Poisson, Joseph-Louis Lagrange, and Carl Gustav Jacob Jacobi; similar operator forms appear in research by Paul Dirac, Hermann Weyl, and Emmy Noether. Generating functions for the sequence connect to combinatorial generating functions exploited by George Pólya, Paul Erdős, and Gian-Carlo Rota; these functions mirror structures found in the work of Leonhard Euler, Jacob Bernoulli, and Augustin-Louis Cauchy.
Recurrence Relations and Differential Equation
Laguerre polynomials satisfy three-term recurrence relations analogous to relations studied by Pafnuty Chebyshev, Chebyshev's school including Sophie Germain, and Chebyshev's colleagues; such recurrences are central in algorithms developed by John von Neumann, Alan Turing, and Donald Knuth. They solve the Laguerre differential equation, a second-order linear ordinary differential equation whose spectral properties relate to Sturm–Liouville theory developed by Jacques Sturm and Joseph Liouville and furthered by Émile Picard and David Hilbert. Stability and asymptotic behavior reference results by G. H. Hardy, E. M. Wright, and Srinivasa Ramanujan.
Special Cases and Normalizations
Several normalizations produce variants used in literature by Felix Klein, Élie Cartan, and Hermann Weyl; physicists such as Wolfgang Pauli, Enrico Fermi, and Paul Dirac used specific normalizations in quantum applications. Special cases include constant and linear polynomials treated in classical texts by Isaac Newton, Gottfried Wilhelm Leibniz, and Leonhard Euler, and normalized forms connect to orthonormal bases constructed in Hilbert space frameworks by David Hilbert, John von Neumann, and Stefan Banach.
Applications in Physics and Numerical Analysis
Laguerre polynomials appear in radial solutions of the hydrogen-like atom treated by Erwin Schrödinger, Paul Dirac, and Niels Bohr, and in quantum harmonic oscillator methods used by Wolfgang Pauli, Enrico Fermi, and Julian Schwinger. Numerical quadrature schemes such as Gauss–Laguerre quadrature were developed in contexts by Carl Friedrich Gauss, Isaac Newton, and James H. Wilkinson and are applied in computations by Grace Hopper, John Backus, and Donald Knuth. Signal processing and spectral methods using these polynomials link to Claude Shannon, Norbert Wiener, and Alan Turing.
Generalizations and Associated Laguerre Polynomials
Associated Laguerre polynomials generalize the classical sequence with an additional parameter; these generalizations are discussed in work by Émile Picard, Jacques Hadamard, and André Weil and are related to hypergeometric functions studied by Carl Friedrich Gauss, Srinivasa Ramanujan, and William Rowan Hamilton. Further extensions connect to orthogonal polynomial families classified by Nikolai Chebyshev, Pafnuty Chebyshev, and Salvatore Pincherle and to special function frameworks developed by George B. Airy, Sir George Gabriel Stokes, and Augustus De Morgan.