hermiteschen

Hermiteschen is a term used in mathematics to denote concepts named after the French mathematician Charles Hermite. The label encompasses several related objects and results that arise from Hermite’s work on polynomials, special functions, and differential equations.

One central family is the Hermite polynomials H_n(x). They form a set of orthogonal polynomials on the real line with respect to the weight function e^{-x^2}. They satisfy the generating function

sum_{n=0}^∞ H_n(x) t^n/n! = e^{2xt - t^2},

and the Rodrigues formula H_n(x) = (-1)^n e^{x^2} d^n/dx^n (e^{-x^2}). The polynomials obey the orthogonality relation ∫_{-∞}^{∞} e^{-x^2} H_m(x) H_n(x) dx = √π 2^n n! δ_{mn}.

Another major component are the Hermite functions, defined by ψ_n(x) = (1/(π^{1/4}√(2^n n!))) e^{-x^2/2} H_n(x). These form an orthonormal basis for the Hilbert space L^2(R). They are eigenfunctions of the Fourier transform (up to a convention-dependent phase), and they play a key role in the quantum harmonic oscillator and in time–frequency analysis.

The Hermite differential equation y'' - 2x y' + 2n y = 0 has polynomial solutions H_n, linking the polynomials to a second-order differential framework. Hermiteschen objects are widely used in approximation theory, probability (through connections to Gaussian distributions and Edgeworth expansions), and physics, particularly in scenarios with Gaussian weights or harmonic potentials. See also Hermite polynomials, Hermite functions, and the Hermite differential equation.