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Hermite Polynomial

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HermiteH

The Hermite polynomials H_n(x) are a set of orthogonal polynomials over the interval (-infty,infty) with weight e^(-x^2) for nonnegative integers n, illustrated above for n=1, 2, 3, and 4. Hermite polynomials are implemented in the Wolfram Language as HermiteH[n, x].

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

H_n(z)=(n!)/(2pii)∮e^(-t^2+2tz)t^(-n-1)dt,
(1)

where the contour has contour winding number 1 around the origin and is traversed counterclockwise (Arfken 1985, p. 416).

The first few Hermite polynomials are

H_0(x)=1
(2)
H_1(x)=2x
(3)
H_2(x)=4x^2-2
(4)
H_3(x)=8x^3-12x
(5)
H_4(x)=16x^4-48x^2+12
(6)
H_5(x)=32x^5-160x^3+120x
(7)
H_6(x)=64x^6-480x^4+720x^2-120
(8)
H_7(x)=128x^7-1344x^5+3360x^3-1680x
(9)
H_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680
(10)
H_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x
(11)
H_(10)(x)=1024x^(10)-23040x^8+161280x^6-403200x^4+302400x^2-30240.
(12)

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 sequence with

g(t)=e^(t^2/4)
(13)
f(t)=1/2t
(14)

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

exp(2xt-t^2)=sum_(n=0)^infty(H_n(x)t^n)/(n!).
(15)

Using a Taylor series shows that

H_n(x)=[(partial/(partialt))^nexp(2xt-t^2)]_(t=0)
(16)
=[e^(x^2)(partial/(partialt))^ne^(-(x-t)^2)]_(t=0).
(17)

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

H_n(x)=(-1)^ne^(x^2)[(partial/(partialx))^ne^(-(x-t)^2)]_(t=0)
(18)
=(-1)^ne^(x^2)(d^n)/(dx^n)e^(-x^2).
(19)

Now define differential operators

O^~_1=-e^(x^2)d/(dx)e^(-x^2)
(20)
O^~_2=e^(x^2/2)(x-d/(dx))e^(-x^2/2).
(21)

It follows that

O^~_1f=-e^(x^2)d/(dx)[fe^(-x^2)]
(22)
=2xf-(df)/(dx)
(23)
O^~_2f=e^(x^2/2)(x-d/(dx))[fe^(-x^2/2)]
(24)
=xf+xf-(df)/(dx)
(25)
=2xf-(df)/(dx),
(26)

so

O^~_1=O^~_2,
(27)

and

-e^(x^2)d/(dx)e^(-x^2)=e^(x^2/2)(x-d/(dx))e^(-x^2/2)
(28)

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

exp(2xt-t^2)=sum_(n=0)^(infty)(H_n(x)t^n)/(n!)
(29)
H_n(x)=(-1)^ne^(x^2)(d^n)/(dx^n)e^(-x^2)
(30)
H_n(x)=e^(x^2/2)(x-d/(dx))^ne^(-x^2/2)
(31)

(Arfken 1985, pp. 712-713 and 720).

For z!=0, the Hermite polynomials may be written as

H_n(z)=(2z)^n_2F_0(-1/2n,-1/2(n-1);;-z^(-2))
(32)
=2^nz^n(z^2)^(-n/2)U(-1/2n,1/2,z^2)
(33)

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

H_n(z)=2^nU(-1/2n,1/2,z^2)
(34)

in the right half-plane Re[z]>0. Complex exponentiation uses the principal branch.

The even and odd Hermite polynomials are related to the associated Laguerre polynomials by

H_(2n)(x)=(-1)^n2^(2n)n!L_n^(-1/2)(x^2)
(35)
H_(2n+1)(x)=(-1)^n2^(2n+1)n!xL_n^(1/2)(x^2),
(36)

where n is a nonnegative integer (Trott 2006). The substitution t=x^2 transforms the weight e^(-x^2) for even polynomials into t^(-1/2)e^(-t) on [0,infty). For odd polynomials, factoring out x gives the weight t^(1/2)e^(-t).

A generating function involving the associated Laguerre polynomials is

sum_(k=0)^inftyL_m^(k-m)(2s^2)H_k(x)(s^k)/(k!)=(s^m)/(m!)H_m(x-2s)e^(-s^2+2xs)
(37)

for nonnegative integer m (Mendas 1993). The series converges for all complex numbers s and x. The case m=0 gives the usual exponential generating function. Expanding L_m^(k-m)(2s^2) by the binomial coefficient formula reduces the general case to derivatives of that exponential generating function.

The Hermite polynomials are related to the derivative of erf by

H_n(z)=1/2(-1)^nsqrt(pi)e^(z^2)(d^(n+1))/(dz^(n+1))erf(z).
(38)

They have a contour integral representation

H_n(x)=(n!)/(2pii)∮e^(-t^2+2tx)t^(-n-1)dt.
(39)

They are orthogonal in the interval (-infty,infty) with respect to the weight e^(-x^2)

int_(-infty)^inftyH_m(x)H_n(x)e^(-x^2)dx=delta_(mn)2^nn!sqrt(pi).
(40)

The Hermite polynomials satisfy the symmetry condition

H_n(-x)=(-1)^nH_n(x).
(41)

For n>=1, they also obey the recurrence relations

H_(n+1)(x)=2xH_n(x)-2nH_(n-1)(x)
(42)
H_n^'(x)=2nH_(n-1)(x).
(43)

By solving the Hermite differential equation, the series

H_(2k)(x)=(-1)^k2^k(2k-1)!![1+sum_(j=1)^(k)((-4k)(-4k+4)...(-4k+4j-4))/((2j)!)x^(2j)]
(44)
=(-2)^k(2k-1)!!_1F_1(-k;1/2;x^2)
(45)
H_(2k+1)(x)=(-1)^k2^(k+1)(2k+1)!![x+sum_(j=1)^(k)((-4k)(-4k+4)...(-4k+4j-4))/((2j+1)!)x^(2j+1)]
(46)
=(-1)^k2^(k+1)(2k+1)!!x_1F_1(-k;3/2;x^2)
(47)

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

(-4k)(-4k+4)...(-4k+4j-4)=4^j(-k)_j,
(48)

with (x)_n the Pochhammer symbol and k a nonnegative integer.

For a>0, let a set of associated functions be defined by

u_n(x)=sqrt(a/(pi^(1/2)n!2^n))H_n(ax)e^(-a^2x^2/2),
(49)

where n is a nonnegative integer. These functions satisfy the orthogonality conditions

int_(-infty)^inftyu_n(x)(du_m)/(dx)dx={asqrt((n+1)/2) m=n+1; -asqrt(n/2) m=n-1; 0 otherwise
(50)
int_(-infty)^inftyu_m(x)u_n(x)dx=delta_(mn)
(51)
int_(-infty)^inftyu_m(x)xu_n(x)dx={1/asqrt((n+1)/2) m=n+1; 1/asqrt(n/2) m=n-1; 0 otherwise
(52)
int_(-infty)^inftyu_m(x)x^2u_n(x)dx={(sqrt(n(n-1)))/(2a^2) m=n-2; (2n+1)/(2a^2) m=n; (sqrt((n+1)(n+2)))/(2a^2) m=n+2; 0 m not in {n-2,n,n+2}
(53)
int_(-infty)^inftye^(-x^2)H_alpha(x)H_beta(x)H_gamma(x)dx=sqrt(pi)(2^salpha!beta!gamma!)/((s-alpha)!(s-beta)!(s-gamma)!)
(54)

for nonnegative integers m, n, alpha, beta, and gamma. The last integral holds 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). For a nonnegative integer r, 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,
(55)

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

The polynomial discriminant is

D_n=2^(3n(n-1)/2)product_(k=1)^nk^k
(56)

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

sum_(k=0)^n(n; k)H_k(x)H_(n-k)(y)=2^(n/2)H_n(2^(-1/2)(x+y))
(57)

and

sum_(k=0)^n(n; k)H_k(x)(2y)^(n-k)=H_n(x+y)
(58)

(G. Colomer, pers. comm.). Another identity is

H_n(x+y)=(H+2y)^n,
(59)

where the umbral calculus convention is H^k=H_k(x) (T. Drane, pers. comm., Feb. 14, 2006).

They also obey the sum

sum_(k=0)^n(-1)^(n-k)(n; k)H_n(k)=2^nn!,
(60)

as well as the more complicated identity

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!),
(61)

where H_n=H_n(0) is a Hermite number, S(n,k) is a Stirling number of the second kind, (x)_n is a Pochhammer symbol, and |_n/2_| uses the floor function (T. Drane, pers. comm., Feb. 14, 2006).

For a positive integer m, a class of generalized Hermite polynomials gamma_n^m(x) satisfying

e^(mxt-t^m)=sum_(n=0)^inftygamma_n^m(x)t^n
(62)

is discussed by Djordjević (1996). A class of related polynomials defined by

h_(n,m)(x)=gamma_n^m((2x)/m)
(63)

and with generating function

e^(2xt-t^m)=sum_(n=0)^inftyh_(n,m)(x)t^n
(64)

was studied by Djordjević (1996). They satisfy

H_n(x)=n!h_(n,2)(x).
(65)

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

Another standard normalization of the Hermite polynomials is defined by

He_n(x)=2^(-n/2)H_n(x/(sqrt(2)))
(66)

(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

He_1(x)=x
(67)
He_2(x)=x^2-1
(68)
He_3(x)=x^3-3x
(69)
He_4(x)=x^4-6x^2+3
(70)
He_5(x)=x^5-10x^3+15x.
(71)

When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 1; -1, 1; -3, 1; 3, -6, 1; 15, -10, 1; ... (OEIS A096713). The polynomial He_n(x) is the matching polynomial of the complete graph K_n for n>=1.

See also

Associated Laguerre Polynomial, Gegenbauer Polynomial, Hermite Number, Jacobi Polynomial, Mehler's Hermite Polynomial Formula, Weber Functions

Related Wolfram sites

https://functions.wolfram.com/Polynomials/HermiteH/, https://functions.wolfram.com/HypergeometricFunctions/HermiteHGeneral/

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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, France, pp. 293-308, 1908.Hermite, C. Oeuvres complètes, tome 3. Paris, France: 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 q-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, Germany: Springer-Verlag, 1948.Mendas, I. "Generating Function for the Product of the Associated Laguerre and Hermite Polynomials." J. Phys. A 26, L93-L95, 1993. https://doi.org/10.1088/0305-4470/26/3/005.National Institute of Standards and Technology. "Definitions." §18.3 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/18.3.National Institute of Standards and Technology. "Interrelations and Limit Relations." §18.7 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/18.7.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.Subramanian, 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.Trott, M. "Relationships Among the Orthogonal Polynomials." §2.9 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Hermite Polynomial

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Weisstein, Eric W. "Hermite Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HermitePolynomial.html

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