Hermite Polynomial

A set of Orthogonal Polynomials. The Hermite polynomials $H_n(x)$ are illustrated above for $x\in[0,1]$ and $n = 1$, 2, ..., 5.

The Generating Function for Hermite polynomials is

$$\mathop{\rm exp}\nolimits (2xt-t^2) \equiv \sum_{n=0}^\infty {H_n(x)t^n\over n!}.$$ (1)

Using a Taylor Series shows that,

$$H_n(x) = \left[{\left({\partial\over \partial t}\right)^n\mathop{\rm exp}\nolimits (2xt-t^2)}\right]_{t=0}$$

$$= \left[{e^{x^2}\left({\partial\over \partial t}\right)^ne^{-(x-t)^2}}\right]_{t=0}.$$ (2)

Since $\partial f(x-t)/\partial t = -\partial f(x-t)/\partial x$,

$$H_n(x) = (-1)^ne^{x^2}\left[{\left({\partial\over \partial x}\right)^ne^{-(x-t)^2}}\right]_{t=0}$$

$$= (-1)^ne^{x^2} {d^n\over dx^n} e^{-x^2}.$$ (3)

Now define operators

$$\tilde O_1 \equiv -e^{x^2} {d\over dx} e^{-x^2}$$ (4)

$$\tilde O_2 \equiv e^{x^2/2}\left({x - {d\over dx}}\right)e^{-x^2/2}.$$ (5)

It follows that

$$\tilde O_1f = -e^{x^2} {d\over dx} [fe^{-x^2}] = 2xf - {df\over dx}$$ (6)

$$\tilde O_2f = e^{x^2/2}\left({x - {d\over dx}}\right)[fe^{-x^2/2}]$$

$$= xf+xf- {df\over dx} = 2xf - {df\over dx},$$ (7)

so

$$\tilde O_1 = \tilde O_2,$$ (8)

and

$$-e^{x^2} {d\over dx} e^{-x^2} = e^{x^2/2}\left({x - {d\over dx}}\right)e^{-x^2/2},$$ (9)

which means the following definitions are equivalent:

$$\mathop{\rm exp}\nolimits (2xt-t^2) \equiv \sum_{n=0}^\infty {H_n(x)t^n\over n!}$$ (10)

$$H_n(x) \equiv (-1)^ne^{x^2} {d^n\over dx^n} e^{-x^2}$$ (11)

$$H_n(x) \equiv e^{x^2/2}\left({x - {d\over dx}}\right)ne^{-x^2/2}.$$ (12)

The Hermite Polynomials are related to the derivative of the Error Function by

$$H_n(z)=(-1)^2 {\sqrt{\pi}\over 2} e^{z^2} {d^{n+1}\over dz^{n+1}} \mathop{\rm erf}\nolimits (z).$$ (13)

They have a contour integral representation

$$H_n(x) = {n!\over 2\pi i} \int e^{-t^2+2tx}t^{-n-1}\,dt.$$ (14)

They are orthogonal in the range $(-\infty, \infty)$ with respect to the Weighting Function $e^{-x^2}$

$$\int_{-\infty}^\infty H_n(x)H_m(x)e^{-x^2}\,dx = \delta_{mn}2^nn!\sqrt{\pi}.$$ (15)

Define the associated functions

$$u_n(x)\equiv\sqrt{a\over \pi^{1/2}n!2^n}\,H_n(ax)e^{-a^2x^2/2}.$$ (16)

These obey the orthogonality conditions

$$\int_{-\infty}^\infty u_n(x){du_m\over dx}\,dx = \left\{\begin{array}{ll} a\sqrt{n+1\over 2} & \mbox{$m=n+1$}\\ -a\sqrt{n\over 2} & \mbox{$m=n-1$}\\ 0 & \mbox{otherwise}\end{array}\right.$$ (17)

$$\int_{-\infty}^\infty u_m(x)u_n(x)\,dx = \delta_{mn}$$ (18)

$$\int_{-\infty}^\infty u_m(x)xu_n(x)\,dx = \left\{\begin{array}{ll} {1\over a}\sqrt{n+!\over 2} & \mbox{$m=n... ...er a}\sqrt{n\over 2} & \mbox{$m=n-1$}\\ 0 & \mbox{otherwise}\end{array}\right.$$ (19)

$$\int_{-\infty}^\infty u_m(x)x^2u_n(x)\,dx = \left\{\begin{array}{ll} {2n+1\over 2a^2} & \mbox{$m=n$}\\ {\sqr... ...2a^2} & \mbox{$m=n+2$}\\ 0 & \mbox{$m\not = n\not = n\pm 2$}\end{array}\right.$$ (20)

$$\int_{-\infty}^\infty e^{-x^2}H_\alpha H_\beta H_\gamma\,dx = \sqrt{\pi} {2^s \alpha!\beta!\gamma!\over (s-\alpha)!(s-\beta)!(s-\gamma)!},$$ (21)

if $\alpha+\beta+\gamma=2s$ is Even and $s\geq \alpha$, $s\geq \beta$, and $s\geq\gamma$. Otherwise, the last integral is 0 (Szegö 1975, p. 390).

They also satisfy the Recurrence Relations

$$H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x)$$ (22)

$$H_n'(x)=2nH_{n-1}(x).$$ (23)

The Discriminant is

$$D_n=2^{3n(n-1)/2} \prod_{\nu=1}^n \nu^\nu$$ (24)

(Szegö 1975, p. 143).

An interesting identity is

$$\sum_{\nu=0}^n{n\choose\nu}H_\nu(x)H_{n-\nu}(y)=2^{n/2}H_n[2^{-1/2}(x+y)].$$ (25)

The first few Polynomials are

$$\begin{eqnarray*} H_0(x) &=& 1\\ H_1(x) &=& 2x\\ H_2(x) &=& 4x^2-2\\ H_3(... ...(x) &=& 1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240. \end{eqnarray*}$$

A class of generalized Hermite Polynomials $\gamma_n^m(x)$ satisfying

$$e^{mxt-t^m}=\sum_{n=0}^\infty \gamma_n^m(x)t^n$$ (26)

was studied by Subramanyan (1990). A class of related Polynomials defined by

$$h_{n,m}=\gamma_n^m\left({2x\over m}\right)$$ (27)

and with Generating Function

$$e^{2xt-t^m}=\sum_{n=0}^\infty h_{n,m}(x)t^n$$ (28)

was studied by Djordjevic (1996). They satisfy

$$H_n(x)=n! h_{n,2}(x).$$ (29)

A modified version of the Hermite Polynomial is sometimes defined by

$$\mathop{\rm He}\nolimits_n(x)\equiv H_n\left({x\over \sqrt{2}}\right).$$ (30)

See also Mehler's Hermite Polynomial Formula, Weber Functions

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Orthogonal Polynomials.'' Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. ``Hermite Functions.'' §13.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 712-721, 1985.

Chebyshev, P. L. ``Sur le développement des fonctions à une seule variable.'' Bull. ph.-math., Acad. Imp. Sc. St. Pétersbourg 1, 193-200, 1859.

Chebyshev, P. L. Oeuvres, Vol. 1. New York: Chelsea, pp. 49-508, 1987.

Djordjevic, G. ``On Some Properties of Generalized Hermite Polynomials.'' Fib. Quart. 34, 2-6, 1996.

Hermite, C. ``Sur un nouveau développement en série de fonctions.'' Compt. Rend. Acad. Sci. Paris 58, 93-100 and 266-273, 1864. Reprinted in Hermite, C. Oeuvres complètes, Vol. 2. Paris, pp. 293-308, 1908.

Hermite, C. Oeuvres complètes, Vol. 3. Paris, p. 432, 1912.

Iyanaga, S. and Kawada, Y. (Eds.). ``Hermite Polynomials.'' Appendix A, Table 20.IV in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1479-1480, 1980.

Sansone, G. ``Expansions in Laguerre and Hermite Series.'' Ch. 4 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 295-385, 1991.

Spanier, J. and Oldham, K. B. ``The Hermite Polynomials $H_n(x)$.'' Ch. 24 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 217-223, 1987.

Subramanyan, P. R. ``Springs of the Hermite Polynomials.'' Fib. Quart. 28, 156-161, 1990.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.