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)