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)
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where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).
The first few Hermite polynomials are
(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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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)
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(14)
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(Roman 1984, p. 30), giving the exponential generating function
(15)
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Using a Taylor series shows that
(16)
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(17)
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Since ,
(18)
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(19)
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Now define operators
(20)
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(21)
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It follows that
(22)
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(23)
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(24)
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(25)
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(26)
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so
(27)
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and
(28)
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(Arfken 1985, p. 720), which means the following definitions are equivalent:
(29)
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(30)
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(31)
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(Arfken 1985, pp. 712-713 and 720).
The Hermite polynomials may be written as
(32)
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(33)
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(Koekoek and Swarttouw 1998), where is a confluent hypergeometric function of the second kind, which can be simplified to
(34)
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in the right half-plane .
The Hermite polynomials are related to the derivative of erf by
(35)
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They have a contour integral representation
(36)
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They are orthogonal in the range with respect to the weighting function
(37)
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The Hermite polynomials satisfy the symmetry condition
(38)
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They also obey the recurrence relations
(39)
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(40)
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By solving the Hermite differential equation, the series
(41)
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(42)
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(43)
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(44)
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are obtained, where the products in the numerators are equal to
(45)
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with the Pochhammer symbol.
Let a set of associated functions be defined by
(46)
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then the satisfy the orthogonality conditions
(47)
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(48)
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(49)
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(50)
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(51)
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if is even and , , and . Otherwise, the last integral is 0 (Szegö 1975, p. 390). Another integral is
(52)
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where and is a binomial coefficient (T. Drane, pers. comm., Feb. 14, 2006).
The polynomial discriminant is
(53)
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(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)
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and
(55)
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(G. Colomer, pers. comm.). A very pretty identity is
(56)
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where (T. Drane, pers. comm., Feb. 14, 2006).
They also obey the sum
(57)
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as well as the more complicated
(58)
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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)
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was studied by Subramanyan (1990). A class of related polynomials defined by
(60)
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and with generating function
(61)
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was studied by Djordjević (1996). They satisfy
(62)
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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)
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(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)
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(65)
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(66)
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(67)
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(68)
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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 .