A set of Orthogonal Polynomials. The Hermite polynomials are illustrated above for and , 2, ..., 5.
The Generating Function for Hermite polynomials is
(1) |
(2) |
(3) |
(4) | |||
(5) |
(6) | |||
(7) |
(8) |
(9) |
(10) | |||
(11) | |||
(12) |
(13) |
(14) |
(15) |
(16) |
These obey the orthogonality conditions
(17) | |||
(18) | |||
(19) | |||
(20) | |||
(21) |
They also satisfy the Recurrence Relations
(22) |
(23) |
The Discriminant is
(24) |
An interesting identity is
(25) |
The first few Polynomials are
A class of generalized Hermite Polynomials satisfying
(26) |
(27) |
(28) |
(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.
© 1996-9 Eric W. Weisstein