Hermite Polynomials

HERMITE_POLYNOMIAL, a C library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

The physicist's Hermite polynomial H(i,x) can be defined by:

        H(i,x) = (-1)^i exp(x^2/2) * d^i/dx^i ( exp(-x^2/2) )

The normalized physicist's Hermite polynomial Hn(i,x) is scaled so that

        Integral ( -oo < x < +oo ) exp ( - x^2 ) * Hn(i,x) Hn(j,x) dx = delta ( i, j )

The probabilist's Hermite polynomial He(i,x) is related to H(i,x) by:

        He(i,x) = H(i,x/sqrt(2)) / sqrt ( 2^in )

The normalized probabilist's Hermite polynomial Hen(i,x) is scaled so that

        Integral ( -oo < x < +oo ) exp ( - 0.5*x^2 ) * Hen(i,x) Hen(j,x) dx = delta ( i, j )

The Hermite function Hf(i,x) is related to H(i,x) by:

        Hf(i,x) = H(i,x) * exp(-x^2/2) / sqrt ( 2^i * i! * sqrt ( pi ) )

The Hermite function Hf(i,x) is scaled so that:

        Integral ( -oo < x < +oo ) Hf(i,x) Hf(j,x) dx = delta ( i, j )


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


HERMITE_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a C library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYOMIAL, a C library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

GEGENBAUER_POLYNOMIAL, a C library which evaluates the Gegenbauer polynomial and associated functions.

HERMITE_EXACTNESS, a C program which tests the polynomial exactness of Gauss-Hermite quadrature rules for estimating the integral of a function with density exp(-x^2) over the interval (-oo,+oo).


HERMITE_PRODUCT_POLYNOMIAL, a C library which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

HERMITE_RULE, a C program which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_POLYNOMIAL, a C library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a C library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a C library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a C library which evaluates the shifted Legendre polynomial, with domain [0,1].

LOBATTO_POLYNOMIAL, a C library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

PCE_BURGERS, a C program which defines and solves a version of the time-dependent viscous Burgers equation, with uncertain viscosity, using a polynomial chaos expansion in terms of Hermite polynomials, by Gianluca Iaccarino.

PCE_ODE_HERMITE, a C program which sets up a simple scalar ODE for exponential decay with an uncertain decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.

POLPAK, a C library which evaluates a variety of mathematical functions.

TEST_VALUES, a C library which supplies test values of various mathematical functions.


  1. Theodore Chihara,
    An Introduction to Orthogonal Polynomials,
    Gordon and Breach, 1978,
    ISBN: 0677041500,
    LC: QA404.5 C44.
  2. Walter Gautschi,
    Orthogonal Polynomials: Computation and Approximation,
    Oxford, 2004,
    ISBN: 0-19-850672-4,
    LC: QA404.5 G3555.
  3. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
    NIST Handbook of Mathematical Functions,
    Cambridge University Press, 2010,
    ISBN: 978-0521192255,
    LC: QA331.N57.
  4. Gabor Szego,
    Orthogonal Polynomials,
    American Mathematical Society, 1992,
    ISBN: 0821810235,
    LC: QA3.A5.v23.

Source Code:

Last revised on 02 July 2019.