Hermite Polynomials

HERMITE_POLYNOMIAL is a FORTRAN77 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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a FORTRAN77 library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a FORTRAN77 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.

HERMITE_EXACTNESS, a FORTRAN77 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 FORTRAN77 library which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

HERMITE_RULE, a FORTRAN77 program which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_POLYNOMIAL, a FORTRAN77 library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Legendre polynomial and associated functions.

PCE_BURGERS, a FORTRAN77 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 FORTRAN77 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 FORTRAN77 library which evaluates a variety of mathematical functions.

TEST_VALUES, a FORTRAN77 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:

Examples and Tests:

List of Routines:

You can go up one level to the FORTRAN77 source codes.

Last revised on 18 October 2014.