释义 |
Hermite PolynomialA set of Orthogonal Polynomials. The Hermite polynomials are illustrated above for and , 2,..., 5.
The Generating Function for Hermite polynomials is
 | (1) |
Using a Taylor Series shows that,
Since ,
Now define operators
It follows that
so
 | (8) |
and
 | (9) |
which means the following definitions are equivalent:
The Hermite Polynomials are related to the derivative of the Error Function by
 | (13) |
They have a contour integral representation
 | (14) |
They are orthogonal in the range with respect to the Weighting Function 
 | (15) |
Define the associated functions
 | (16) |
These obey the orthogonality conditions
if is Even and , , and . Otherwise, thelast integral is 0 (Szegö 1975, p. 390).
They also satisfy the Recurrence Relations
 | (22) |
 | (23) |
The Discriminant is
 | (24) |
(Szegö 1975, p. 143).
An interesting identity is
 | (25) |
The first few Polynomials are
A class of generalized Hermite Polynomials satisfying
 | (26) |
was studied by Subramanyan (1990). A class of related Polynomials defined by
 | (27) |
and with Generating Function
 | (28) |
was studied by Djordjevic (1996). They satisfy
 | (29) |
A modified version of the Hermite Polynomial is sometimes defined by
 | (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 .'' 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. |