The sequence that is inverse to the one similarly denoted but without the minus sign, and thus speak of Hermite polynomials of negative variance.
12.
In the case of the harmonic potential, the wave functions solutions of the one-dimensional quantum harmonic oscillator are known as Hermite polynomials.
13.
The interested reader may investigate other functional linear operators which give rise to different kinds of orthogonal eigenfunctions such as Legendre polynomials, Chebyshev polynomials and Hermite polynomials.
14.
If the notation " He " is used for these Hermite polynomials, and " H " for those above, then these may be characterized by
15.
Accurate description of such a beam involves expansion over some complete, orthogonal set of functions ( over two-dimensions ) such as Hermite polynomials or the Ince polynomials.
16.
For example, the quantum harmonic oscillator is ideally expanded in Hermite polynomials, and Jacobi-polynomials can be used to define the associated Legendre functions typically appearing in rotational problems.
17.
Where " x " i = 0 for all " i " > 2; thus allowing for a combinatorial interpretation of the coefficients of the Hermite polynomials.
18.
Among the most notable Appell sequences besides the trivial example { " x " " n " } are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials.
19.
The authors Kemp and Kemp have called it " Hermite distribution " from the fact its probability function and the moment generating function can be expressed in terms of the coefficients of ( modified ) Hermite polynomials.
20.
Neither article in Wikipedia states such a relationship but there is a link to an external article on Hermite interpolation from the article on Hermite polynomials . . . but that could just have been places in error.
