Hermite Polynomial

A 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,
 (2)

Since ,
 (3)

Now define operators
 (4) (5)

It follows that
 (6) (7)

so
 (8)

and
 (9)

which means the following definitions are equivalent:
 (10) (11) (12)

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

 (17) (18) (19) (20) (21)

if is Even and , , and . Otherwise, the last 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)

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.