Finite series with factorials (legendre polynomials for x=1)

[bbking22]

Hi there. I study legendre polynomials Pn(x) and I have a problem : I can't prove that Pn(1)=1.

(M=n/2)
I have try many things. I need help. Thank you.

 

[Count Iblis]

Re: Finite series with factorials (legendre polynomials for

Hi there. I study legendre polynomials Pn(x) and I have a problem : I can't prove that Pn(1)=1.

(M=n/2)
I have try many things. I need help. Thank you.

It's easier using the recurrence relation. Verify it for n = 0 (trivial) and n = 1 and then use induction. If you have to use the given formula, you could still derive the recurrence formula from it first and use the recurrence relation anyway

 

[bbking22]

Thank you for your help and your response.

The given form of the Legendre polynomials comes from the solution of the Legendre equation after we multiply with (2n)!/(2^n*(n!)^2) (quantity that gives the solution the form of Legendre polynomials as we know them) in a way that gave me the feeling that it was some kind of identity. That's why I tried the binomial expansion of 1=1^(n/2)=(2-1)^(n/2) but with no luck. I have try some other similar ways but again with no luck.

I can see that the method you suggest works if you know what is to be proved. My problem is that I still can't understand how it was first proved. I mean, if you don't know that Pn(1)=1 how you can find the value of Pn(1) and how can you force (by mulitplying with the right quantity) Pn(1) to be unity?

Thank you again!

 

[Denis]

Are you Riley “B.B.” King, the great blues singer and guitarist :?:

 

[Count Iblis]

Thank you for your help and your responce.
In fact the given form of the legendre polynomials comes from the solution of the Legendre equation after we multiply with (2n)!/(2^n*(n!)^2) (quantity that gives the solution the form of Legendre polynomials as we know them) in a way that gave me the feeling that it was some kind of identity. That's why i tried the binomial expansion of 1=1^(n/2)=(2-1)^(n/2) but with no luck. I have try some other similar ways but again with no luck.
I can see that the method you suggest works if you know what is to be proved. My problem is that I still can't understand how it was first proved. I mean, if you don't know that Pn(1)=1 how you can find the value of Pn(1) and how can you force (by mulitplying with the right quantity) Pn(1) to be unity?
Thank you again!

Many identities involving such binomial summations are often first noticed for some special cases and then proved by induction or other means. Why not read the book A=B, it contains many examples of binomial summations and techniques on how to prove identities