Hermite polynomials

Chebyshev–Hermite polynomials

Polynomials orthogonal on $ ( - \infty , \infty ) $ with the weight function $ h ( x) = e ^ {- x ^ {2} } $. The standardized Hermite polynomials are defined by the Rodrigues formula

$$ H _ {n} ( x) = ( - 1 ) ^ {n} e ^ {x ^ {2} } ( e ^ {- x ^ {2} } ) ^ {( n)} . $$

The most commonly used formulas are:

$$ H _ {n+} 1 ( x) = 2 x H _ {n} ( x) - 2 n H _ {n- 1} ( x) , $$

$$ H _ {n} ^ \prime ( x) = 2 n H _ {n- 1} ( x) , $$

$$ H _ {n} ( x) = \sum _ {k=0} ^ { [ n/2] } \frac{(- 1) ^ {k} n ! }{k ! ( n - 2 k ) ! } ( 2 x ) ^ {n-2k} , $$

$$ \mathop{\rm exp} ( 2 x w - w ^ {2} ) = \sum _ {n=0}^ \infty \frac{H _ {n} ( x) }{n!} w ^ {n} . $$

The first few Hermite polynomials are:

$$ H _ {0} ( x) = 1,\ H _ {1} ( x) = 2x,\ H _ {2} ( x) = 4 x ^ {2} - 2 , $$

$$ H _ {3} ( x) = 8 x ^ {3} - 12 x ,\ H _ {4} ( x) = 16 x ^ {4} - 48 x ^ {2} + 12 , $$

$$ H _ {5} ( x) = 32 x ^ {5} + 160 x ^ {3} + 120 x ,\dots . $$

The polynomial $ H _ {n} ( x) $ satisfies the differential equation

$$ y ^ {\prime\prime} - 2 x y ^ \prime + 2 n y = 0 . $$

The orthonormal Hermite polynomials are defined by

$$ \widehat{H} _ {n} ( x) = \frac{H _ {n} ( x) }{\sqrt {n ! 2 ^ {n} \sqrt \pi } } . $$

The Hermite polynomials with leading coefficient one have the form

$$ \widetilde{H} _ {n} ( x) = \frac{1}{2 ^ {n} } H _ {n} ( x) = \ \frac{(- 1) ^ {n} }{2 ^ {n} } e ^ {x ^ {2} } ( e ^ {- x ^ {2} } ) ^ {(n) }. $$

Fourier series in Hermite polynomials in the interior of $ ( - \infty , \infty ) $ behave analogous to trigonometric Fourier series.

In mathematical statistics and probability theory one uses the Hermite polynomials corresponding to the weight function

$$ h ( x) = \mathop{\rm exp} ( - x ^ {2} / 2 ) . $$

The definition of Hermite polynomials is encountered in P. Laplace [1]. A detailed study of them was published by P.L. Chebyshev in 1859 (see [2]). Later, these polynomials were studied by Ch. Hermite . V.A. Steklov [4] proved that the set of them is dense in the space of square-summable functions with the weight $ h ( x) = \mathop{\rm exp} ( - x ^ {2} ) $ on the whole real line.

References

[1]	P.S. Laplace, Mém. Cl. Sci. Math. Phys. Inst. France , 58 (1810) pp. 279–347
[2]	P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 335–341 (In Russian)
[3a]	Ch. Hermite, C.R. Acad. Sci. Paris , 58 (1864) pp. 93–100
[3b]	Ch. Hermite, C.R. Acad. Sci. Paris , 58 (1864) pp. 266–273
[4]	V.A. Steklov, Izv. Akad. Nauk , 10 (1956) pp. 403–416
[5]	P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1979) (In Russian)

Comments

The result by Steklov mentioned in the last sentence of the main article goes back at least to H. Weyl (1908), cf. the references in [a3], Sect. 5.7.

One possible way to prove the Plancherel formula for the Fourier transform is by use of Hermite polynomials, cf. [a4]. Hermite polynomials occur in solutions of the heat and Schrödinger equations and in the so-called heat polynomials, cf. [a1]. A canonical orthonormal basis of the representation space for the Schrödinger representation of the Heisenberg group is given in terms of Hermite polynomials, cf. [a2].

References

[a1]	W. Miller jr., "Symmetry and separation of variables" , Addison-Wesley (1977)
[a2]	W. Schempp, "Harmonic analysis on the Heisenberg nilpotent Lie group" , Longman (1986)
[a3]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[a4]	E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)