A question on the orthogonal polynomial

In summary, the speaker is asking for help with integrating Laguerre polynomials and has found three useful equations on a math function website. They are having trouble with the third equation and are asking for assistance or recommendations for learning how to derive the desired integration formula. Another person suggests looking into Dixon's identity and the rules for the Gamma function, which helps the speaker solve the problem.
  • #1
sufive
23
0
Dear All Friends,

I am currently working on a project which needs some orthogonality
integration formulae of Laguerre polynomials. I referred worlfram's math
function site
http://functions.wolfram.com/Polynomials/LaguerreL3/21/02/01/
and get three seemingly useful ones.

However, as very natural exercises, when I made tests by setting
(p=1,alpha=lambda+1,beta=lambda,m=n) in the third equation of the
above web-page and try to get the first one, I always cannot accomplish
the goal. So I suspect the inter-consistances of the three integration formulae
in the above web-page.

Is here someone professional in orthogonality polynomials and would like
to resolve my suspicious? Or can here someone recommend me some
materials so that I can learn and derive out the desired integration formulae
by myself?

Thank you very much!
 
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  • #2
Check out Dixon's identity and the rules for the Gamma function (basically that it's just factorial for integer arguments) and you will find your way from eq.3 to eq.1
 
  • #3
Thank you very much! I worked it out just as you suggested!
 

1. What are orthogonal polynomials?

Orthogonal polynomials are a set of polynomials that are mutually perpendicular in a given interval with respect to a specific weight function. They are often used in mathematical and statistical analysis because of their unique properties.

2. What is the purpose of orthogonal polynomials?

The main purpose of orthogonal polynomials is to simplify complex mathematical expressions and calculations. They can be used to approximate functions, solve differential equations, and perform numerical integration, among other applications.

3. How are orthogonal polynomials different from regular polynomials?

Orthogonal polynomials have the additional property of being mutually perpendicular, while regular polynomials do not have this property. This means that they are orthogonal to each other when integrated over a specific interval with respect to a chosen weight function.

4. What is the significance of the weight function in orthogonal polynomials?

The weight function in orthogonal polynomials plays a crucial role in determining the properties and behavior of the polynomials. It is usually chosen based on the specific application or problem at hand, and can greatly influence the accuracy and efficiency of using orthogonal polynomials.

5. Can orthogonal polynomials be used in real-world applications?

Yes, orthogonal polynomials have many practical applications in fields such as physics, engineering, and statistics. They are particularly useful in solving differential equations, approximating functions, and performing numerical integration. They also have applications in signal processing, data analysis, and computer graphics.

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