Legendre polynomials and Rodrigues' formula

In summary, the conversation discusses the second order differential equation for fn, which is (1-x^2)f''_n-2(n-1)xf'_n+2nf_n=0. The author then applies the General Leibniz formula and obtains (1-x^2)f^{(n+2)}_n-2xf^{(n+1)}_n+n(n+1)f^{(n)}_n=0, which is Legendre's differential equation. This equation is satisfied by the normalized Legendre polynomials Pn(x). However, the equation given in the text, y=2nn! for x=1, is incorrect and attempting to normalize the Legendre polynomial with x=1
  • #1
TimeRip496
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Source: http://www.nbi.dk/~polesen/borel/node4.html#1
Differentiating this equation we get the second order differential eq. for fn,
[itex](1-x^2)f''_n-2(n-1)xf'_n+2nf_n=0[/itex] ....(22)
But when I differentiate to 2nd order, I get this instead,
[itex](1-x^2)f''_n+2(n-1)xf'_n+2nf_n=0[/itex]Applying General Leibniz formula to (22) we easily get,
[itex](1-x^2)f^{(n+2)}_n-2xf^{(n+1)}_n+n(n+1)f^{(n)}_n=0[/itex] ,...(24)
How did the author get to (24)? I thought he get that by subbing the Leibniz formula into (22) by replacing the f with fn but it doesn't get to (24) and it doesn't make sense at all.
What I get,
[itex](1-x^2)f^{(n+2)}_n+2(n-1)xf^{(n+1)}_n+2nf^{(n)}_n=0[/itex]

which is exactly Legendre's differential equation (1-49). This equation is therefore satisfied by the polynomials
$$y=\frac{d^n}{dx^n}(x^2-1)^n$$......(25)
The Legendre polynomials Pn(x) are normalized by the requirement Pn(1)=1. Using y=2nn! for x=1,
Is equation (25) supposed to be the Legendre polynomial? And why do we normalized with x=1?Besides, attempt to normalize the Legendre polynomial with x=1 doesn't get me anywhere.
$$P_l(1)=\sum^n_{k=0} (-1)^k \frac{(2n-2k)!}{2^nk!(n-k)!(n-2k)!}*1$$
 
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  • #2
Your initial statement, that the text is wrong, is correct. I haven't gone further.
 

1. What are Legendre polynomials?

Legendre polynomials are a set of orthogonal polynomials that are commonly used in mathematical physics and engineering. They are named after the French mathematician Adrien-Marie Legendre, who first introduced them in 1782.

2. What is the significance of Legendre polynomials?

Legendre polynomials have many important applications, including solving partial differential equations, expressing multivariate functions, and approximating other functions. They are also used in numerical analysis and in the field of quantum mechanics.

3. What is Rodrigues' formula?

Rodrigues' formula is a method for generating Legendre polynomials using a single formula. It was discovered by the French mathematician Olinde Rodrigues in the 1800s. The formula involves using derivatives and factorials to express the polynomials in terms of their degree and variable.

4. How are Legendre polynomials related to spherical harmonics?

Legendre polynomials are the building blocks for spherical harmonics, which are used to represent solutions to the Laplace's equation in spherical coordinates. This relationship is important in studying physical phenomena such as heat flow and electrostatics in spherical systems.

5. Can Legendre polynomials be generalized to higher dimensions?

Yes, Legendre polynomials can be extended to higher dimensions using the concept of hyperspherical harmonics. These are analogous to spherical harmonics in three dimensions and are used in solving differential equations in higher dimensions.

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