# Product of Legendre Polynomials

• Repetit
In summary, the person is asking for help understanding how to express the product of two Legendre polynomials as a sum of Legendre polynomials using the recursion formula. They also provide an example of what they want to achieve and a source for further information.
Repetit
Hey!

Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula

$$(l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0$$

but I am not sure how to do this. What is basically want to do is

$$P_n(x) P_m(x) = \sum\limits_i c_i P_i(x)$$

I hope my question is understandable.

Thanks!

Repetit said:
Hey!

Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula

$$(l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0$$

but I am not sure how to do this. What is basically want to do is

$$P_n(x) P_m(x) = \sum\limits_i c_i P_i(x)$$

I hope my question is understandable.

Thanks!

"http://www.mscand.dk/article.php?id=1633" "

Last edited by a moderator:

## What is the definition of a Product of Legendre Polynomials?

The Product of Legendre Polynomials is a mathematical expression that results from multiplying two or more Legendre polynomials together. Legendre polynomials are a special type of orthogonal polynomial that are often used in physics and engineering to describe certain physical phenomena.

## How do you calculate a Product of Legendre Polynomials?

To calculate a Product of Legendre Polynomials, you first need to multiply the coefficients of the individual polynomials. Then, you can use the recurrence relation or the Rodrigues formula to expand the product into a sum of Legendre polynomials. Finally, you can use the orthogonality property of Legendre polynomials to simplify the expression further.

## What is the significance of Product of Legendre Polynomials in physics?

The Product of Legendre Polynomials is often used to describe the angular dependence of solutions to certain types of differential equations in physics, such as the Schrödinger equation. This product allows for the representation of complicated functions in terms of simpler Legendre polynomials, making it a useful tool in many physical applications.

## What are the properties of Product of Legendre Polynomials?

The Product of Legendre Polynomials inherits many properties from the individual polynomials that make up the product. Some of these properties include orthogonality, completeness, and recurrence relations. Additionally, the product has the property of being a polynomial of order n+m, where n and m are the orders of the individual polynomials being multiplied.

## What are some real-world applications of Product of Legendre Polynomials?

The Product of Legendre Polynomials has many applications in physics, engineering, and mathematics. Some examples include the study of quantum mechanics, solving Laplace's equation in spherical coordinates, and approximating solutions to differential equations. It is also used in signal processing, image analysis, and other fields where orthogonal polynomials are useful in representing complex data.

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