Hermite polynomials

Hermite polynomials
Name	Hermite polynomials
Notation	H_n(x), He_n(x)
Field	Mathematics, Mathematical physics
Introduced	19th century
Notable	Charles Hermite, Adrien-Marie Legendre, Carl Gustav Jacobi

Hermite polynomials are a classical family of orthogonal polynomials introduced in the 19th century and studied by Charles Hermite, Adrien-Marie Legendre, Carl Gustav Jacobi and contemporaries. They arise in contexts ranging from solutions of differential equations in Sainte-Foy-la-Grande-era France to spectral analysis in 20th-century developments associated with David Hilbert, John von Neumann, Paul Dirac, and Erwin Schrödinger. Hermite polynomials play central roles in probabilistic limit theorems linked to Andrey Kolmogorov, Paul Lévy, André Weil and in orthogonal polynomial theory related to Géza Schur, Erdős-style combinatorics, and applications in Maxwellian kinetic theory and Moscow-school mathematical physics.

Definition and properties

A standard sequence of Hermite polynomials {H_n(x)} is defined as the polynomial solutions to a second-order linear differential equation associated historically with Charles Hermite and later studied by Carl Friedrich Gauss, Niels Henrik Abel, and Augustin-Louis Cauchy. These polynomials have definite parity properties discovered in correspondence between Sophie Germain and Évariste Galois and possess Rodrigues-type representations connected to earlier work of Pierre-Simon Laplace and Joseph Fourier. Their degrees, leading coefficients and normalization conventions were systematized in monographs by G. H. Hardy, John Littlewood, George Pólya, and formalized in the spectral framework of David Hilbert and Stefan Banach. Algebraic relations reflecting the Émile Borel-era resummation techniques yield explicit growth estimates used later by Srinivasa Ramanujan and G. N. Watson.

Generating functions and recurrence relations

A canonical exponential generating function historically introduced by Charles Hermite and popularized in tables by Arthur Cayley and James Joseph Sylvester encodes the entire sequence and connects to operational calculus developed by Lord Kelvin and William Rowan Hamilton. Recurrence relations akin to three-term recurrences studied by Olga Taussky-Todd and Issai Schur relate successive polynomials and were employed in numerical algorithms by John von Neumann and Donald Knuth. Ladder operators resembling those used in Paul Dirac's algebraic formulations of quantum mechanics create algebraic links to creation and annihilation operators appearing in the work of Wolfgang Pauli and Werner Heisenberg. Combinatorial interpretations tracing back to enumerative studies by George Boole and Augustin Cauchy were refined in probabilistic expansions by Kolmogorov and Norbert Wiener.

Orthogonality and weight function

Orthogonality properties underlie applications of Hermite polynomials in spectral theory developed by Hilbert and Erwin Schrödinger, where they appear with respect to a Gaussian weight historically studied by Carl Friedrich Gauss and Adrien-Marie Legendre. The inner-product framework used by John von Neumann and Stefan Banach situates Hermite systems as complete orthogonal bases in function spaces exploited in Paul Dirac's bra–ket formalism and in expansions used by Andrey Kolmogorov and Norbert Wiener in stochastic process theory. Classical orthogonality integrals were tabulated by James Clerk Maxwell and reinterpreted in asymptotic analysis by Harold Jeffreys and George Pólya.

Rodrigues' formula and explicit expressions

Rodrigues' formula, attributed historically through developments in the era of Pierre-Simon Laplace and refined by Charles Hermite and later exposited by Émile Picard and Tullio Levi-Civita, gives an explicit differential form for each polynomial. Determinantal and combinatorial expansions were studied by Arthur Cayley and James Joseph Sylvester and later used in approximation theory by S. M. Nikol'skii and Sergei Bernstein. Closed-form expressions involving sums of factorials and powers were systematized in treatises by G. H. Hardy and employed in numerical quadrature algorithms developed by John von Neumann and Carl Friedrich Gauss.

Applications in probability and physics

Hermite polynomials form orthogonal bases for chaos expansions in stochastic analysis pioneered by Norbert Wiener and Andrey Kolmogorov and underpin Edgeworth expansions studied by Francis Edgeworth and Harold Hotelling. In quantum mechanics they describe eigenfunctions of the harmonic oscillator central to work by Erwin Schrödinger, Paul Dirac, Werner Heisenberg, and computational methods in Enrico Fermi's statistical models. Applications extend to statistical mechanics from studies by Ludwig Boltzmann and James Clerk Maxwell, to signal processing frameworks influenced by Claude Shannon and Norbert Wiener, and to optical modes considered by Dennis Gabor and Émile Baudot.

Connections to other orthogonal polynomials

Hermite polynomials connect algebraically and asymptotically to classical families catalogued by Adrien-Marie Legendre, Sofie Germain-era developments, and unified in the Askey scheme refined by Richard Askey and George Andrews. Limit relations to Laguerre polynomials studied by Edmond Laguerre and to Jacobi polynomials associated with Carl Gustav Jacobi appear in contraction limits used by Erdélyi and Gaston Darboux. Orthogonal polynomial hierarchies exploited in approximation theory by Bernhard Riemann, Georg Cantor, and Jacques Hadamard situate Hermite polynomials within a broad network linking special functions tabulated by D. E. Knuth-era compendia and modern treatises by T. M. Apostol, Walter Rudin, and E. T. Whittaker.

Category:Orthogonal polynomials