How are the Hermite Polynomials Defined and Calculated?

[DreamWeaver]

Defining the Hermite Polynomials by:

\(\displaystyle \mathscr{H}_n(x) = (-1)^n e^{x^2}\, \frac{d^n}{dx^n} \Bigg\{ e^{-x^2}\Bigg\}\)Show that

\(\displaystyle \mathscr{H}_5(x) = 32 x^5-160 x^3+120 x\)

 

[lfdahl]

Spoiler

\[H_5 = -e^{x^2}\cdot \frac{\mathrm{d^5} }{\mathrm{d} x^5}\left \{ e^{-x^2} \right \} =-e^{x^2}\cdot \frac{\mathrm{d^4} }{\mathrm{d} x^4}\left \{ -2xe^{-x^2} \right \}\\\\ =-e^{x^2}\cdot \frac{\mathrm{d^3} }{\mathrm{d} x^3}\left \{ (4x^2-2)e^{-x^2} \right \} =-e^{x^2}\cdot \frac{\mathrm{d^2} }{\mathrm{d} x^2}\left \{ (-8x^3+12x)e^{-x^2} \right \}\\\\ =-e^{x^2}\cdot \frac{\mathrm{d} }{\mathrm{d} x}\left \{ (16x^4-48x^2+12)e^{-x^2} \right \} =-e^{x^2}\cdot \left \{ (-32x^5+160x^3-120x)e^{-x^2} \right \}\\\\ =32x^5-160x^3+120x\]

 

[chisigma]

Defining the Hermite Polynomials by:

\(\displaystyle \mathscr{H}_n(x) = (-1)^n e^{x^2}\, \frac{d^n}{dx^n} \Bigg\{ e^{-x^2}\Bigg\}\)Show that

\(\displaystyle \mathscr{H}_5(x) = 32 x^5-160 x^3+120 x\)

[sp]The recursive relation for Hermite Polynomials is...

$\displaystyle H_{n+1}(x) = 2\ x\ H_{n}(x) - 2\ n\ H_{n-1} (x)\ (1)$

... and is...

$\displaystyle H_{0}= 1$

$\displaystyle H_{1} = 2\ x$

... so that...

$\displaystyle H_{2} = 4\ x^{2} - 2$

$\displaystyle H_{3} = 8\ x^{3} - 12\ x$

$\displaystyle H_{4} = 16\ x^{4} - 48\ x^{2} + 12$

$\displaystyle H_{5} = 32\ x^{5} - 160\ x^{3} + 120\ x$

[/sp]

Kind regards

$\chi$ $\sigma$

 

[DreamWeaver]

Spoiler

\[H_5 = -e^{x^2}\cdot \frac{\mathrm{d^5} }{\mathrm{d} x^5}\left \{ e^{-x^2} \right \} =-e^{x^2}\cdot \frac{\mathrm{d^4} }{\mathrm{d} x^4}\left \{ -2xe^{-x^2} \right \}\\\\ =-e^{x^2}\cdot \frac{\mathrm{d^3} }{\mathrm{d} x^3}\left \{ (4x^2-2)e^{-x^2} \right \} =-e^{x^2}\cdot \frac{\mathrm{d^2} }{\mathrm{d} x^2}\left \{ (-8x^3+12x)e^{-x^2} \right \}\\\\ =-e^{x^2}\cdot \frac{\mathrm{d} }{\mathrm{d} x}\left \{ (16x^4-48x^2+12)e^{-x^2} \right \} =-e^{x^2}\cdot \left \{ (-32x^5+160x^3-120x)e^{-x^2} \right \}\\\\ =32x^5-160x^3+120x\]

[sp]The recursive relation for Hermite Polynomials is...

$\displaystyle H_{n+1}(x) = 2\ x\ H_{n}(x) - 2\ n\ H_{n-1} (x)\ (1)$

... and is...

$\displaystyle H_{0}= 1$

$\displaystyle H_{1} = 2\ x$

... so that...

$\displaystyle H_{2} = 4\ x^{2} - 2$

$\displaystyle H_{3} = 8\ x^{3} - 12\ x$

$\displaystyle H_{4} = 16\ x^{4} - 48\ x^{2} + 12$

$\displaystyle H_{5} = 32\ x^{5} - 160\ x^{3} + 120\ x$

[/sp]

Kind regards

$\chi$ $\sigma$

Two excellent, elegant, and - even better - different proofs. Thanks for taking part, people! (Sun)

Gethin

 

[CellCoree]

The Hermite polynomials are a family of polynomials that arise in the study of mathematical physics, specifically in the field of quantum mechanics. They are named after the French mathematician Charles Hermite, who first defined them in the 19th century.

The differentiation problem mentioned in the content refers to the method used to define the Hermite polynomials. The formula given in the content is known as the generating function for Hermite polynomials. It relates the polynomials to the exponential function, making it easier to calculate their values.

To show that \mathscr{H}_5(x) = 32 x^5-160 x^3+120 x, we can use the differentiation property of the generating function. According to this property, the n-th derivative of the generating function is equal to n times the corresponding Hermite polynomial. In other words,

\frac{d^n}{dx^n} \Bigg\{ e^{-x^2}\Bigg\} = n \mathscr{H}_n(x)

Substituting n = 5 in the above formula, we get

\frac{d^5}{dx^5} \Bigg\{ e^{-x^2}\Bigg\} = 5 \mathscr{H}_5(x)

To calculate the fifth derivative, we can use the product rule and the chain rule. The result is

\frac{d^5}{dx^5} \Bigg\{ e^{-x^2}\Bigg\} = 16x^5 - 120x^3 + 120x

Equating this to 5 times \mathscr{H}_5(x), we get

5\mathscr{H}_5(x) = 16x^5 - 120x^3 + 120x

Dividing both sides by 5, we get the desired result

\mathscr{H}_5(x) = 32 x^5-160 x^3+120 x

Hence, we have successfully shown that the fifth Hermite polynomial can be expressed as 32x^5-160x^3+120x, which is in accordance with the formula given in the content. This method of defining the Hermite polynomials using the generating function is a powerful tool for calculating their values and has numerous applications in physics and engineering.