Integral involving Hermite polynomials

In summary, The conversation discusses the calculation of the integral of H_n(x) * H_m(x) * exp(-c^2 x^2) with x going from -infinity to +infinity and c differs from unity. The speaker is looking for a general formula for this integral and suggests trying specific values for n and m to see if a pattern emerges. However, it is stated that there is no known formula for this integral.
  • #1
Heirot
151
0
Hello!

Is there any way of calculating the integral of H_n(x) * H_m(x) * exp(-c^2 x^2) with x going from -infinity to +infinity and c differs from unity. I'm aware that c=1 is trivial case of orthogonality but I'm really having a problem with the general case. (I should say that this isn't a homework assignment, rather curiosity).

Any ideas?

Thanks...
 
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  • #2
I don't believe that there is any nice formula for that. However, notice that you can calculate that explicitly if you fix some small indexes n and m. It could be, that if you do that, then some pattern becomes visible. So why not calculate it with (n,m)=(1,1), (0,2), (1,3), (2,2), (2,4) and see what happens?
 

What is an integral involving Hermite polynomials?

An integral involving Hermite polynomials is an equation that involves the integration of a function that includes Hermite polynomials as its main component. The Hermite polynomials are a set of orthogonal polynomials that are commonly used in mathematical analysis and physics.

What is the significance of Hermite polynomials in integrals?

Hermite polynomials are significant in integrals because they can be used to approximate many different types of functions. They have important applications in quantum mechanics and statistical mechanics, and are also used in the calculation of Gaussian integrals.

How are Hermite polynomials related to Gaussian distributions?

Hermite polynomials are closely related to Gaussian distributions. In fact, they are the basis functions for the expansion of a Gaussian distribution in a series. This means that any Gaussian distribution can be expressed as a sum of Hermite polynomials, making them a powerful tool in statistical analysis.

What is the general formula for an integral involving Hermite polynomials?

The general formula for an integral involving Hermite polynomials is ∫Hn(x)e-x2dx = (n-1)Hn-1(x)-(n-1)∫Hn-2(x)e-x2dx, where Hn(x) denotes the nth Hermite polynomial. This formula can be derived using the recurrence relation for Hermite polynomials.

What are some common applications of integrals involving Hermite polynomials?

Integrals involving Hermite polynomials have many applications in physics, engineering, and mathematics. They are used in the calculation of quantum mechanical wave functions, the analysis of Brownian motion, and the solution of differential equations. They are also used in the numerical evaluation of complicated integrals and in the development of efficient computational algorithms.

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