# Legendre polynomials proof question.Help

• aligator123
Then, after you've differentiated, you can set t=1 to get the formula for Pn'(1)you mean at the first i have to change n to n-1?Yes, in the sum on the right hand side, replace every n with n-1. So the first term would be P_{n-1}(x). I'm sorry, i didn't understand what you meant exactly. I am going to bed now, so i will look at this again tomorrow. I know of another way to do this problem, hopefully i will have time to explain it to you tomorrow.Edit: Before i go, i will show you what i mean by what i said above. I
aligator123
Hello everyone i had some questions about legendre polynomials. I have solved most of them but i had just two not answered question. I tried to solve this problem by rodriguez rule but it was really hard for me. Could anyone help me or give me some hints for this question?

http://img139.imageshack.us/img139/7731/mathkw4.th.jpg

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Have you tried to solve it using the generating function identity?

The first one is really easy. Just use the ODE whose fundamental solutions are Legendre's polynomials of the first & second kind.

Daniel.

For the second you'll need to perform this integral

$$\int_{0}^{\pi} \left(\cos t \right)^{2n} \ dt \ , \ n\in \mathbb{N}$$

Daniel.

could you give me detailed solution for understanding it better.

sorry but i really got troubled with these questions.could anyone give me detail suggestions?

Hello why no one give any hints?
i tried also generating function.but i couldn't find.
can anyone give me suggestion or show me a way to solve these problems?

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Use the generatling function. Expand g=[1+2tx+t^2]^{-1} in the binomial series. For the first case, use dg/dx for x=1.
For the second case, use dg/dx for x=0.

i tried generating function but i couldn't solve it.could you tell me step by step to solve this?

aligator123 said:
i tried generating function but i couldn't solve it.could you tell me step by step to solve this?

How about you post and tell us where you are stuck?

Remember, the homework helpers here will help you solve the problem, not solve the problems for you.

ok how can i send you my solution?

You can post it in this thread using LaTeX.
Here's a tutorial on how it's used. You can click on any LaTeX image and see the code for the image.

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For the first one, it looks like you've guessed the final result, rather than proven it.
As Meir Achuz told you, use the generating function

$$\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^{\infty} P_n(x) t^n$$

So for the first one, differentiate wrt x, set x=1 and equate coefficients of t^n (using the binomial series expansion). Use a similar procedure for the second question.

what does it meant wrt x?

wrt x?

any help??

aligator123 said:
any help??

wrt= short for "with respect to". As for help, i think you've gotten enough ideas to solve the problem.

I c a certain reluctance towards accepting advice. Maybe you shouldn't have come here in the first place. I hope that no one on this board will help someone by solving the problem entirely, because, to me, that's what you're asking for...

Daniel.

primitive polynomials in GF(4)

no i don't want anyone to do my homework.i want to do myself.
but when i derivate the left side wrt x

2t/(1-2tx+t^2)=d/dx EPn(x)t^n

end then how can i equate? or am i doing sometihng wrong

aligator123 said:
no i don't want anyone to do my homework.i want to do myself.
but when i derivate the left side wrt x

2t/(1-2tx+t^2)=d/dx EPn(x)t^n
The LHS (left hand side) should be t/(1-2tx+t^2)^{3/2}.
Then set x=1 on both sides,and us the binomial expansion. You will get a binomial coefficient that happens to equal n(n+1)/2.

1/(1-2xt+t^2)=sum Pn(x)t^n
d/dx1/(1-2xt+t^2)=d/dx sum Pn(x)t^n
t/(1-2xt+t^2)^3/2=Sum t^nP'_n(x).

and then i tried to equate but i couldn't get the answer
of
P'n(1)=n(n+1)/2

1/(1-2xt+t^2)=sum Pn(x)t^n
d/dx1/(1-2xt+t^2)=d/dx sum Pn(x)t^n
t*(1-2xt+t^2)^-3/2=SumP'n(x)t^n.

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(1-u)^-1/2=sum $$\left( \begin{array}{c} -1/2 \\ n \end{array} \right)$$((-1)^n)(u^n)

t*(1-2xt+t^2)^-3/2=t(1-t)^-3

u=t

(1-u)^-3=sum $$\left( \begin{array}{c} -3 \\ n \end{array} \right)$$((-1)^n)(t^n)

$$\left( \begin{array}{c} -3 \\ n \end{array} \right)$$=(-1)^n*$$\left( \begin{array}{c} n+2 \\ n \end{array} \right)$$

am i right?
but after these calculation i couldn't find

Pn'(1)=n(n+1)/2

at the last equation this equation remains

Pn'(1)=t*(n+2)(n+1)/2

any suggestions

need suggestions?

aligator123 said:
at the last equation this equation remains

Pn'(1)=t*(n+2)(n+1)/2
The t can't be there. You have to change n in the original sum to n-1.

## 1. What are Legendre polynomials?

Legendre polynomials are a set of orthogonal polynomials that are commonly used in mathematics and physics. They are named after the French mathematician Adrien-Marie Legendre and are often denoted by Pn(x), where n is the degree of the polynomial.

## 2. What is the proof for Legendre polynomials?

The proof for Legendre polynomials involves using the Gram-Schmidt process to construct a set of orthogonal polynomials from a given set of polynomials. This process involves finding an orthogonal basis for a given vector space, which in this case is the space of polynomials.

## 3. Why are Legendre polynomials important?

Legendre polynomials are important because they have many applications in mathematics and physics. They are used to solve differential equations, approximate functions, and represent physical phenomena such as electromagnetic fields and quantum mechanical wave functions.

## 4. How do Legendre polynomials relate to spherical harmonics?

Legendre polynomials are closely related to spherical harmonics, which are used to represent solutions to the Laplace equation in spherical coordinates. The Legendre polynomials form the radial part of the spherical harmonics, while the angular part is given by the associated Legendre polynomials.

## 5. Are there any alternative proofs for Legendre polynomials?

Yes, there are alternative proofs for Legendre polynomials, such as using generating functions or using properties of the Gamma function. However, the Gram-Schmidt process is the most commonly used and intuitive method for proving Legendre polynomials.

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