How to Express the Product of Two Legendre Polynomials?

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SUMMARY

The product of two Legendre polynomials, denoted as P_n(x) and P_m(x), can be expressed as a sum of Legendre polynomials using the recursion formula: (l+1)P_{l+1}(x) - (2l+1)xP_l(x) + lP_{l-1}(x) = 0. This formula allows for the transformation of the product into the desired sum form: P_n(x) P_m(x) = ∑_i c_i P_i(x). Understanding this relationship is crucial for applications in mathematical physics and engineering.

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  • Familiarity with Legendre polynomials
  • Understanding of recursion formulas in polynomial mathematics
  • Basic knowledge of mathematical series and summation
  • Experience with mathematical notation and expressions
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  • Study the properties of Legendre polynomials and their applications in physics
  • Learn about the orthogonality of Legendre polynomials
  • Explore the derivation of the recursion formula for Legendre polynomials
  • Investigate the use of generating functions for Legendre polynomials
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Repetit
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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!
 
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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!


For people searching an answer

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