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Hermite Polynomial

The Hermite polynomials are set of orthogonal polynomials over the domain with weighting function , illustrated above for , 2, 3, and 4. Hermite polynomials are implemented in the Wolfram Language as HermiteH[n, x].

The Hermite polynomial 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

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(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 may be called Hermite numbers.

The Hermite polynomials are a Sheffer sequence with

			(13)
			(14)

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

(15)

Using a Taylor series shows that

			(16)
			(17)

Since ,

			(18)
			(19)

Now define operators

			(20)
			(21)

It follows that

			(22)
			(23)
			(24)
			(25)
			(26)

(27)

and

(28)

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

			(29)
			(30)
			(31)

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

The Hermite polynomials may be written as

			(32)
			(33)

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

(34)

in the right half-plane .

The Hermite polynomials are related to the derivative of erf by

(35)

They have a contour integral representation

(36)

They are orthogonal in the range with respect to the weighting function

(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

			(41)
			(42)
			(43)
			(44)

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

(45)

with the Pochhammer symbol.

Let a set of associated functions be defined by

(46)

then the satisfy the orthogonality conditions

			(47)
			(48)
			(49)
		${(sqrt(n(n-1)))/(2a^2) m=n-2; (2n+1)/(2a^2) m=n; (sqrt((n+1)(n+2)))/(2a^2) m=n+2; 0 m!=n!=n+/-2$	(50)
			(51)

if is even and , , and . 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 and 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 , , , , ... (OEIS A054373).

Two interesting identities involving are given by

(54)

and

(55)

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

(56)

where (T. Drane, pers. comm., Feb. 14, 2006).

They also obey the sum

(57)

as well as the more complicated

(58)

where is a Hermite number, is a Stirling number of the second kind, and is a Pochhammer symbol (T. Drane, pers. comm., Feb. 14, 2006).

A class of generalized Hermite polynomials satisfying

(59)

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

			(64)
			(65)
			(66)
			(67)
			(68)

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 is the independence polynomial of the complete graph .

Hermite Polynomial

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