# hermite_polynomial

hermite_polynomial, an Octave 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 )
```

### Languages:

hermite_polynomial is available in a C version and a C++ version and a Fortran90 version and a Fortran77 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

hermite_rule, an Octave code which computes and prints a Gauss-Hermite quadrature rule.

octave_polynomial, an Octave code which analyzes a variety of polynomial families, returning the polynomial values, coefficients, derivatives, integrals, roots, or other information.

pce_ode_hermite, an Octave code 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, an Octave code which evaluates a variety of mathematical functions.

polynomial_conversion, an Octave code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Lagrange, Laguerre and other forms.

test_values, an Octave code which supplies test values of various mathematical functions.

### 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.