## `orthpoly::hermite` -- the Hermite polynomials

#### Introduction

`orthpoly::hermite``(n,x)` computes the value of the n-th degree Hermite polynomial at the point x.

#### Call(s)

`orthpoly::hermite(n, x)`

#### Parameters

`n` | - | a nonnegative integer: the degree of the polynomial.
`x` | - | an indeterminate or an arithmetical expression. An indeterminate is either an identifier (of domain type `DOM_IDENT`) or an indexed identifier (of type `"_index"`).

#### Returns

If `x` is an indeterminate, then a polynomial of domain type `DOM_POLY` is returned. If `x` is an arithmetical expression, then the value of the Hermite polynomial at this point is returned as an arithmetical expression. If `n` is not a nonnegative integer, then `orthpoly::hermite` returns itself symbolically.

#### Details

- These polynomials have integer coefficients.

#### Example 1

Polynomials of domain type `DOM_POLY` are returned, if identifiers or indexed identifiers are specified:

    >> orthpoly::hermite(2, x)

                                        2
                                poly(4 x  - 2, [x])

    >> orthpoly::hermite(3, x[1])

                                     3
                          poly(8 x[1]  - 12 x[1], [x[1]])

However, using arithmetical expressions as input the ``values'' of these polynomials are returned:

    >> orthpoly::hermite(2, 6*x)

                                         2
                                    144 x  - 2

    >> orthpoly::hermite(3, x[1] + 2)

            2 (x[1] + 2) (2 (x[1] + 2) (2 x[1] + 4) - 2) - 8 x[1] - 16

``Arithmetical expressions'' include numbers:

    >> orthpoly::hermite(2, sqrt(2)), orthpoly::hermite(3, 8 + I),
       orthpoly::hermite(1000, 0.3);

                         6, 3808 + 1516 I, 2.26821486e1433

If no integer degree is specified, then `orthpoly::hermite` returns itself symbolically:

    >> orthpoly::hermite(n, x), orthpoly::hermite(1/2, x)

                orthpoly::hermite(n, x), orthpoly::hermite(1/2, x)

#### Background

- The Hermite polynomials are given by the recursion formula

      H(n,x)=2*x*H(n-1,x)-2*(n-1)*H(n-2,x)

  with H(0,x)=1 and H(1,x)=2*x.
- These polynomials are orthogonal on the real line with respect to the weight function w(x)=exp(-x^2).

#### Changes

- Indexed identifiers are now regarded as indeterminates. Non-integer degrees n are now accepted, leading to an unevaluated function call. Efficiency was improved.

Do you have questions or comments?
Copyright © SciFace Software GmbH & Co. KG 2000