Polynomial of Hermit

In Mathematical, the polynomials of Hermit are a continuation of Polynôme S which were named thus in the honor of Charles Hermite. They are defined as follows:

$H\_n\; (X)\; =\; (-\; 1)\; ^n\; e^\; \{x^2/2\}\; \backslash \; frac\; \{d^n\}\; \{dx^n\}\; e^\; \{-\; x^2/2\}$ (form known as probabilistic )

$\backslash \; hat\; H\_n\; (X)\; =\; (-\; 1)\; ^n\; e^\; \{x^2\}\; \backslash \; frac\; \{d^n\}\; \{dx^n\}\; e^\; \{-\; x^2\}$ (form known as physical )

The two definitions are bound by the property of following scale: $\backslash \; hat\; H\_n\; (X)\; =\; 2^\; \{n/2\}\; H\_n\; (\backslash \; sqrt\; \{2\}\; \backslash ,\; X)\; \backslash ,\; \backslash !$.

The first polynomials of Hermit are the following:

$H\_0\; (X)\; =1~$

$H\_1\; (X)\; =x~$

$H\_2\; (X)\; =x^2-1~$

$H\_3\; (X)\; =x^3-3x~$

$H\_4\; (X)\; =x^4-6x^2+3~$

$H\_5\; (X)\; =x^5-10x^3+15x~$

$H\_6\; (X)\; =x^6-15x^4+45x^2-15~$

$\backslash \; hat\; H\_0\; (X)\; =1~$

$\backslash \; hat\; H\_1\; (X)\; =2x~$

$\backslash \; hat\; H\_2\; (X)\; =4x^2-2~$

$\backslash \; hat\; H\_3\; (X)\; =8x^3-12x~$

$\backslash \; hat\; H\_4\; (X)\; =16x^4-48x^2+12~$

$\backslash \; hat\; H\_5\; (X)\; =32x^5-160x^3+120x~$

$\backslash \; hat\; H\_6\; (X)\; =64x^6-480x^4+720x^2-120~$

One can show that in $\{H\_p\}\; ~$ the coefficients of order having the same parity that $p-1~$ is null and that the coefficients of order $p~$ and $p-2~$ are worth respectively $1~$ and $-p\; (p-1)\; /2~$.

Orthogonality

$H\_n$ is a polynomial of $n$ degree. These polynomials are orthogonal for the measurement $\backslash \; mu$ of density

$\backslash \; frac\; \{D\; \backslash \; driven\; (X)\}\{dx\}\; =\; \backslash \; frac\; \{e^\; \{-\; x^2/2\}\}\; \{\backslash \; sqrt\; \{2\; \backslash \; pi\}\}.$

They check:

$\backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; H\_n\; (X)\; H\_m\; (X)\; \backslash ,\; e^\; \{-\; x^2/2\}\; \backslash ,\; dx=n!\; \backslash \; sqrt\; \{2\; \backslash \; pi\}\; ~\; \backslash \; delta\_\; \{Nm\}$

where $\backslash \; delta\_\; \{Nm\}$ is the Symbole of Kronecker. These functions thus form an orthogonal base of the Espace of Hilbert $L\_2\; (\backslash \; mathbb\; C,\; \backslash \; driven)$ of the functions boréliennes such as:

in which the scalar Produit is given by the integral

$\backslash \; langle\; F,\; G\; \backslash \; rangle=\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (X)\; \backslash \; overline\; \{G\; (X)\}\backslash ,\; \backslash \; frac\; \{e^\; \{-\; x^2/2\}\}\; \{\backslash \; sqrt\; \{2\; \backslash \; pi\}\}\; \backslash ,\; dx.$

Similar properties are verifiable for the polynomials of Hermit under their physical shape.

Various properties

The $n$-ième polynomial of Hermit satisfies the differential equation following (in its two versions probabilist or physics):

$H\_n$ (X) - xH_n' (X) +nH_n (X) =0. \,

$\backslash \; hat\; H\_n$ (X) - 2x \ hat H_n' (X) +2n \ hat H_n (X) =0. \,

The polynomials satisfy the property

$H\_n\text{'}\; (X)\; =nH\_\; \{n-1\}\; (X),\; \backslash ,$

$\backslash \; hat\; H\_n\text{'}\; (X)\; =\; 2n\; \backslash \; hat\; H\_\; \{n-1\}\; (X),\; \backslash ,$

that one can write thus

$H\_n\; (x+y)\; =\; \backslash \; sum\_\; \{k=0\}\; ^n\; \{N\; \backslash \; choose\; K\}\; x^k\; H\_\; \{n-k\}\; (there)$

$\backslash \; hat\; H\_n\; (x+y)\; =\; \backslash \; sum\_\; \{k=0\}\; ^n\; \{N\; \backslash \; choose\; K\}\; (2x)\; ^k\; \backslash \; hat\; H\_\; \{n-k\}\; (there)$

They thus check the relation of following recurrence:

$H\_\; \{n+1\}\; (X)\; =\; xH\_\; \{N\}\; (X)\; -\; nH\_\; \{n-1\}\; (X),\; \backslash ,$

$\backslash \; hat\; H\_\; \{n+1\}\; (X)\; =\; 2x\; \backslash \; hat\; H\_\; \{N\}\; (X)\; -\; 2n\; \backslash \; hat\; H\_\; \{n-1\}\; (X).\; \backslash ,$

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