Hermite polynomials

[3] Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new.

Noting from the outset that there are two different standardizations in common use, one convenient method is as follows: These equations have the form of a Rodrigues' formula and can also be written as,

is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.

The Hermite polynomials (probabilist's or physicist's) form an orthogonal basis of the Hilbert space of functions satisfying

The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind.

An explicit formula of Hermite polynomials in terms of contour integrals (Courant & Hilbert 1989) is also possible.

The corresponding expressions for the physicist's Hermite polynomials H follow directly by properly scaling this:[7]

This equality is valid for all complex values of x and t, and can be obtained by writing the Taylor expansion at x of the entire function z → e−z2 (in the physicist's case).

One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as

one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

If X is a random variable with a normal distribution with standard deviation 1 and expected value μ, then

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:

Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments:

This expansion is needed to resolve the wavefunction of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle.

A finer approximation,[9] which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution

The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions:

Similar to Taylor expansion, some functions are expressible as an infinite sum of Hermite polynomials.

There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish.

The existence of some formal power series g(D) with nonzero constant coefficient, such that Hen(x) = g(D)xn, is another equivalent to the statement that these polynomials form an Appell sequence.

The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is

This binomial type identity, for α = β = ⁠1/2⁠, has already been encountered in the above section on #Recursion relations.)

the sequence that is inverse to the one similarly denoted, but without the minus sign, and thus speak of Hermite polynomials of negative variance.

This formula can be used in connection with the recurrence relations for Hen and ψn to calculate any derivative of the Hermite functions efficiently.

For real x, the Hermite functions satisfy the following bound due to Harald Cramér[14][15] and Jack Indritz:[16]

The Hermite functions ψn(x) are thus an orthonormal basis of L2(R), which diagonalizes the Fourier transform operator.

This is a fundamental result for the quantum harmonic oscillator discovered by Hip Groenewold in 1946 in his PhD thesis.

Equivalently, it is the number of involutions of an n-element set with precisely k fixed points, or in other words, the number of matchings in the complete graph on n vertices that leave k vertices uncovered (indeed, the Hermite polynomials are the matching polynomials of these graphs).

where δ is the Dirac delta function, ψn the Hermite functions, and δ(x − y) represents the Lebesgue measure on the line y = x in R2, normalized so that its projection on the horizontal axis is the usual Lebesgue measure.

and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution