How can I show the expansion of Hermite Polynomials using exponential functions?

In summary, the conversation discusses the expansion of e^{-y^2+2xy} using hermite polynomials. The speaker has tried to manipulate the expansion but has encountered an issue with the polynomial satisfying only one condition instead of the required three. They ask for help and mention the two other conditions that the polynomials must satisfy.
  • #1
MathematicalPhysicist
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I need to show that:
[tex]\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}y^n=e^{-y^2+2xy}[/tex]
where H_n(x) is hermite polynomial.

Now I tried the next expansion:
[tex]e^{-y^2}e^{2xy}=\sum_{n=0}^{\infty}\frac{(-y)^{2n}}{n!}\cdot \sum_{k=0}^{\infty}\frac{(2xy)^k}{k!}[/tex]
after some simple algebraic rearrangemnets i got:
[tex]\sum_{n=0}^{\infty}(2x-y)^n\frac{y^n}{n!}[/tex]
which looks similar to what i need to show, the problem is that the polynomial (2x-y)^n satisifes only the condition: H_n'(x)=2nH_n-1(x)
and not the other two conditions, so i guess something is missing, can anyone help me on this?

thanks in advance.
 
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  • #2
[tex]e^{-y^2}=\sum_{n=0}^{\infty}\frac{(-y)^{2n}}{n!}[/tex]
Is this correct?

I'm inclined to think that:
[tex]e^{-y^2}=\sum_{n=0}^{\infty}\frac{(-y^2)^{n}}{n!}[/tex]
[tex]=\sum_{n=0}^{\infty}\frac{(-1)^n y^{2n}}{n!}[/tex].?
 
  • #3
yes, ofcourse you are correct, i mixed between them, can you please help me on this?
 
  • #4
What two other conditions do these polynomials have to satisfy?
 
  • #5
H''_n-2xH'_n+2nH_n=0
H_n+1-2xH_n+2nH_n-1=0

according to this exercise.

well I looked at wikipedia, and I guess I only need to use the first definition given at wikipedia.
 

1. What are Hermite Polynomials?

Hermite Polynomials are a type of mathematical function that are used to solve various problems in physics and engineering. They are named after the French mathematician Charles Hermite, who first studied them in the 19th century.

2. What are the properties of Hermite Polynomials?

Hermite Polynomials have many important properties, including being orthogonal, meaning that they are perpendicular to each other when graphed. They also have unique recurrence relations and generate the set of all polynomials.

3. How are Hermite Polynomials used in physics?

Hermite Polynomials are commonly used in physics to solve problems related to quantum mechanics, such as finding the wavefunction of a harmonic oscillator. They are also used in statistical mechanics to describe the energy distribution of particles in a system.

4. Can Hermite Polynomials be generalized to higher dimensions?

Yes, Hermite Polynomials can be extended to higher dimensions. These are known as Hermite Multivariate Polynomials and are used to solve problems in areas such as multivariate statistics and signal processing.

5. What are some real-world applications of Hermite Polynomials?

Hermite Polynomials have many practical applications in fields such as physics, engineering, and statistics. They are used in areas such as signal processing, image analysis, and data compression. They are also used in the calculation of probabilities in statistics and finance.

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