# hermite_polynomial

hermite_polynomial, a MATLAB 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 MATLAB version.

### Related Data and Programs:

bernstein_polynomial, a MATLAB code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

chebyshev_polynomial, a MATLAB code 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 MATLAB code which evaluates the Gegenbauer polynomial and associated functions.

gen_hermite_rule, a MATLAB code which can compute and print a generalized Gauss-Hermite quadrature rule.

hermite_product_display, a MATLAB code which displays an image of a function created by the Cartesian product of two Hermite polynomials, such as f(x,y) = h(3,x) * h(1,y).

hermite_product_polynomial, a MATLAB code which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

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

jacobi_polynomial, a MATLAB code which evaluates the Jacobi polynomial and associated functions.

laguerre_polynomial, a MATLAB code which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

legendre_polynomial, a MATLAB code which evaluates the Legendre polynomial and associated functions.

lobatto_polynomial, a MATLAB code which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

pce_ode_hermite, a MATLAB 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, a MATLAB code which evaluates a variety of mathematical functions.

test_values, a MATLAB 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.