# hermite_polynomial

hermite_polynomial, a C++ code 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 )

### Licensing:

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

### Languages:

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++ code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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PCE_BURGERS, a C++ code 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.

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### Reference:

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 16 March 2020