Relation between Hermite and associated Laguerre

In summary, the conversation is about proving the expression H_{2n}(x)=(-1)^n2^{2n}n!L_n^{-\frac{1}{2}}(x^2), involving the Hermite and associated Laguerre polynomials. The person asking for help is advised to provide some effort or thoughts on the problem before seeking assistance. If they are completely stuck, they should ask for a hint to get started.
  • #1
Suvadip
74
0
Please help me in proving the following expression

\(\displaystyle H_{2n}(x)=(-1)^n2^{2n}n!L_n^{-\frac{1}{2}}(x^2)\)

where \(\displaystyle H_n\) is the Hermite polynomial and \(\displaystyle L_n^{-\frac{1}{2}}\) is the associated Laguerre polynomial.
 
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  • #2
Hello suvadip,

By now you should know we expect some effort to be given, such as the work you have tried, or the thoughts you have on how to proceed, for the reasons I have already given. You may already have tried something, and one of our helpers may give you information you already know, and this would be a waste of the helper's time, which is valuable.

If you simply have no idea how to begin, then you should state this, and ask for a hint to begin.
 

FAQ: Relation between Hermite and associated Laguerre

What is the relation between Hermite and associated Laguerre polynomials?

The Hermite and associated Laguerre polynomials are two families of orthogonal polynomials that are closely related. The associated Laguerre polynomials can be obtained from the Hermite polynomials by a simple change of variables. Specifically, the associated Laguerre polynomial Lnk(x) can be expressed in terms of the Hermite polynomial Hn+k(x) as Lnk(x) = (-1)k/n! Hn+k(x).

What are the properties of Hermite and associated Laguerre polynomials?

Both the Hermite and associated Laguerre polynomials have numerous important properties. Some of the key properties include: orthogonality with respect to specific weight functions, recurrence relations, differential equations satisfied by the polynomials, generating functions, and explicit formulas for the coefficients. These properties make these polynomials useful in many areas of mathematics and physics.

How are Hermite and associated Laguerre polynomials used in physics?

The Hermite and associated Laguerre polynomials are widely used in physics, particularly in the study of quantum mechanics. They arise in the solution of the Schrödinger equation for various physical systems, such as the harmonic oscillator and the hydrogen atom. They also play a key role in the study of statistical mechanics and quantum field theory.

What is the significance of the zeros of Hermite and associated Laguerre polynomials?

The zeros of both the Hermite and associated Laguerre polynomials have important implications in mathematics and physics. For instance, the zeros of the Hermite polynomials are closely related to the eigenvalues of the harmonic oscillator, while the zeros of the associated Laguerre polynomials are related to the energy levels of the hydrogen atom. In addition, the distribution of the zeros of these polynomials has been extensively studied and has connections to topics such as random matrix theory and number theory.

Are there any applications of Hermite and associated Laguerre polynomials in other fields?

Yes, the Hermite and associated Laguerre polynomials have applications in various fields outside of mathematics and physics. For example, they are used in signal processing, probability theory, and computer science. In addition, they have been used in the development of efficient numerical algorithms for solving differential equations and other mathematical problems.

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