Hermites

Hermites is a term used in mathematics to denote several concepts named after the French mathematician Charles Hermite (1822–1901). The name appears in analysis, algebra, and numerical methods, reflecting Hermite’s influence on the theory of functions, polynomials, and matrix forms.

One of the best known uses is Hermite polynomials. There are two common families: the physicists’ Hermite polynomials Hn(x) and the probabilists’ Hermite polynomials He,n(x). The polynomials satisfy recurrence relations such as Hn+1(x) = 2x Hn(x) − 2n Hn−1(x) with H0 = 1 and H1 = 2x, and He,0(x) = 1, He,1(x) = x, with He,n+1(x) = x He,n(x) − n He,n−1(x). They arise as eigenfunctions of the quantum harmonic oscillator (physicists’ version) and as orthogonal polynomials with Gaussian weights in both contexts. Hermite polynomials play a role in probability, spectral theory, and numerical analysis, including approximation and solving differential equations.

Hermite interpolation is another related concept. It is a form of polynomial interpolation that matches not only function values at selected nodes but also specified derivatives, yielding smoother approximations in numerical analysis.

Hermite normal form is a canonical form for integer matrices under unimodular row operations. An integer matrix can be transformed into an upper triangular form with certain divisibility properties, providing a tool for solving linear Diophantine systems and for computing Smith normal forms.

Historically, many of these constructs honor Charles Hermite’s contributions to algebra, elliptic functions, and the theory of mathematical functions. In addition to the named polynomials, Hermite’s influence extends to related functions and methods used across analysis and algebra.