Hermite Polynomial (original) (raw)

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HermiteH

The Hermite polynomials H_n(x) are set of orthogonal polynomials over the domain (-infty,infty) with weighting function e^(-x^2) , illustrated above for n=1 , 2, 3, and 4. Hermite polynomials are implemented in theWolfram Language as HermiteH[n,_x_].

The Hermite polynomial H_n(z) can be defined by the contour integral

	(1)

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).

The first few Hermite polynomials are

When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 2; -2, 4; -12, 8; 12, -48, 16; 120, -160, 32; ... (OEIS A059343).

The values H_n(0) may be called Hermite numbers.

The Hermite polynomials are a Sheffer sequencewith

(Roman 1984, p. 30), giving the exponential generating function

	(15)

Using a Taylor series shows that

Since partialf(x-t)/partialt=-partialf(x-t)/partialx ,

Now define operators

It follows that

	(27)

and

	(28)

(Arfken 1985, p. 720), which means the following definitions are equivalent:

(Arfken 1985, pp. 712-713 and 720).

The Hermite polynomials may be written as

(Koekoek and Swarttouw 1998), where U(a,b,z) is a confluent hypergeometric function of the second kind, which can be simplified to

	(34)

in the right half-plane R[z]>0 .

The Hermite polynomials are related to the derivative of erfby

	(35)

They have a contour integral representation

	(36)

They are orthogonal in the range (-infty,infty) with respect to the weighting function e^(-x^2)

	(37)

The Hermite polynomials satisfy the symmetry condition

	(38)

They also obey the recurrence relations

	(39)

	(40)

By solving the Hermite differential equation, the series

are obtained, where the products in the numerators are equal to

	(45)

with (x)_n the Pochhammer symbol.

Let a set of associated functions be defined by

	(46)

then the u_n satisfy the orthogonality conditions

if alpha+beta+gamma=2s is even and s>=alpha , s>=beta , and s>=gamma . Otherwise, the last integral is 0 (Szegö 1975, p. 390). Another integral is

$int_(-infty)^inftyu_n(x)x^ru_m(x)dx={0 if r-n-m is odd; (r!)/((2a)^r)sqrt((2^(m+n))/(m!n!))sum_(p=max(0,-s))^(min(m,n))(n; p)(m; p)(p!)/(2^p(s+p)!) otherwise,$	(52)

where s=(r-n-m)/2 and (n; k) is a binomial coefficient (T. Drane, pers. comm., Feb. 14, 2006).

The polynomial discriminant is

	(53)

(Szegö 1975, p. 143), a normalized form of the hyperfactorial, the first few values of which are 1, 32, 55296, 7247757312, 92771293593600000, ... (OEIS A054374). The table of resultants is given by {0} , {-8,0} , {0,-2048,0} , {192,16384,28311552,0} , ... (OEIS A054373).

Two interesting identities involving H_n(x+y) are given by

	(54)

and

	(55)

(G. Colomer, pers. comm.). A very pretty identity is

	(56)

where H^k=H_k(x) (T. Drane, pers. comm., Feb. 14, 2006).

They also obey the sum

	(57)

as well as the more complicated

| $H_n(x)=H_n+sum_(m=0)^(\|_n/2_|)[sum_(k=1)^(n-2m)(-1)^kS(n-2m,k)(-x)_k]×((-1)^m2^(n-2m)n!)/((n-2m)!m!),$ | (58) | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---- |

where H_n=H_n(0) is a Hermite number, S(n,k) is a Stirling number of the second kind, and (x)_n is a Pochhammer symbol (T. Drane, pers. comm., Feb. 14, 2006).

A class of generalized Hermite polynomials gamma_n^m(x) satisfying

	(59)

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

	(60)

and with generating function

	(61)

was studied by Djordjević (1996). They satisfy

	(62)

Roman (1984, pp. 87-93) defines a generalized Hermite polynomial H_n^((nu))(x) with variance .

A modified version of the Hermite polynomial is sometimes (but rarely) defined by

	(63)

(Jörgensen 1916; Magnus and Oberhettinger 1948; Slater 1960, p. 99; Abramowitz and Stegun 1972, p. 778). The first few of these polynomials are given by

When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 1; , 1; , 1; 3, , 1; 15, , 1; ... (OEIS A096713). The polynomial He_n(x) is the independence polynomial of the complete graph K_n .

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References

Abramowitz, M. and Stegun, I. 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.Andrews, G. E.; Askey, R.; and Roy, R. "Hermite Polynomials." §6.1 in Special Functions. Cambridge, England: Cambridge University Press, pp. 278-282, 1999.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.Djordjević, 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, tome 2. Paris, pp. 293-308, 1908.Hermite, C. Oeuvres complètes, tome 3. Paris: Hermann, 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.Jeffreys, H. and Jeffreys, B. S. "The Parabolic Cylinder, Hermite, and Hh Functions" §23.08 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 620-622, 1988.Jörgensen, N. R. Undersögler over frekvensflader og korrelation. Copenhagen, Denmark: Busck, 1916.Koekoek, R. and Swarttouw, R. F. "Hermite." §1.13 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its -Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 50-51, 1998.Magnus, W. and Oberhettinger, F. Ch. 5 in Formeln und Sätze für die speziellen Funktionen der mathematischen Physik, 2nd ed. Berlin: Springer-Verlag, 1948.Roman, S. "The Hermite Polynomials." §4.2.1 in The Umbral Calculus. New York: Academic Press, pp. 30 and 87-93, 1984.Rota, G.-C.; Kahaner, D.; Odlyzko, A. "Hermite Polynomials." §10 in "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.Sansone, G. "Expansions in Laguerre and Hermite Series." Ch. 4 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 295-385, 1991.Slater, L. J. Confluent Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1960.Sloane, N. J. A. Sequences A054373,A054374, A059343, and A096713 in "The On-Line Encyclopedia of Integer Sequences."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.