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# Legendre and Recurrence relations Watch

1. If someone could make sense of this or send a good link to a proof, you'd be a god send.

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2. (Original post by DoYouEvenMaths)
If someone could make sense of this or send a good link to a proof, you'd be a god send.

are you aware of the generating function for Legendre Polynomials?
3. (Original post by TeeEm)
are you aware of the generating function for Legendre Polynomials?
no
4. (Original post by DoYouEvenMaths)
no
the only proof I know uses the generating function

5. (Original post by TeeEm)
the only proof I know uses the generating function

I don't believe this proof needs the generating functions (it only uses them to derive the recurrences that are given here).

It seems a slightly strange homework - it's a pretty huge ask to expect someone to find the solution without googling, but if you google it's basically a "copy out some notes" question.
6. (Original post by DFranklin)
I don't believe this proof needs the generating functions (it only uses them to derive the recurrences that are given here).

It seems a slightly strange homework - it's a pretty huge ask to expect someone to find the solution without googling, but if you google it's basically a "copy out some notes" question.

I am sure you are right is just that the proof I vaguely remember starts from there.

I do not quite remember this staff, but what is stuck in my mind is that these special functions (such as Legendre and Bessel) have many results and relationships and proving something sometimes is a matter what you take as a given.
7. (Original post by TeeEm)
I do not quite remember this staff, but what is stuck in my mind is that these special functions (such as Legendre and Bessel) have many results and relationships and proving something sometimes is a matter what you take as a given.
Yes absolutely. I would think most people first see the ODE, because it comes up "naturally" when you deal with PDEs using cylindrical (or is it spherical? whatever...) coordinates. And then you derive some recurrence relations. Which is basically 100% the other direction from what's being asked here.
8. (Original post by DFranklin)
Yes absolutely. I would think most people first see the ODE, because it comes up "naturally" when you deal with PDEs using cylindrical (or is it spherical? whatever...) coordinates. And then you derive some recurrence relations. Which is basically 100% the other direction from what's being asked here.
agreed
9. (Original post by DFranklin)
I don't believe this proof needs the generating functions (it only uses them to derive the recurrences that are given here).

It seems a slightly strange homework - it's a pretty huge ask to expect someone to find the solution without googling, but if you google it's basically a "copy out some notes" question.
I haven't done this sort of thing "in anger" for over 25 years, but I thought I'd have a go just to see if the old magic was still there as I used to enjoy this sort of methods-type question in my braver days

Apart from one false start and a stupid inability to replace n with n+1 consistently I managed to get it out reasonably quickly, but I guess you need to pick the right strategy otherwise it's easy to go round in circles!

I started by differentiating the 1st eq w.r.t.x and then using the 2nd eq to substitute for

This gives in terms of and .

Multiplying this new eq by x and comparing with the original 2nd eq with n replaced by n+1 then gives in terms of and .

Differentiation w.r.t.x gives an equation in and and the latter can be substituted using one of our earlier results.

A bit of simple algebra then gives the required result
10. (Original post by davros)
I haven't done this sort of thing "in anger" for over 25 years, but I thought I'd have a go just to see if the old magic was still there as I used to enjoy this sort of methods-type question in my braver days

Apart from one false start and a stupid inability to replace n with n+1 consistently I managed to get it out reasonably quickly, but I guess you need to pick the right strategy otherwise it's easy to go round in circles!

I started by differentiating the 1st eq w.r.t.x and then using the 2nd eq to substitute for

This gives in terms of and .

Multiplying this new eq by x and comparing with the original 2nd eq with n replaced by n+1 then gives in terms of and .

Differentiation w.r.t.x gives an equation in and and the latter can be substituted using one of our earlier results.

A bit of simple algebra then gives the required result

Maybe in the summer when I retire properly I might revisit some of this material to keep the "old cogs" turning.
11. (Original post by TeeEm)

Maybe in the summer when I retire properly I might revisit some of this material to keep the "old cogs" turning.
A bit sad, but when I was a sixth former I used to find differential equations fascinating, and because I was getting interested in things like quantum mechanics I'd fetch out some advanced books from the local library and make an effort to try to prove some of the relations for Legendre, Laguerre, Hermite and Chebyshev polynomials just from the basic definitions given. It didn't require a lot of true insight but gave me a lot of confidence in the sort of extended manipulations used in later Methods courses.
12. (Original post by davros)
A bit sad, but when I was a sixth former I used to find differential equations fascinating, and because I was getting interested in things like quantum mechanics I'd fetch out some advanced books from the local library and make an effort to try to prove some of the relations for Legendre, Laguerre, Hermite and Chebyshev polynomials just from the basic definitions given. It didn't require a lot of true insight but gave me a lot of confidence in the sort of extended manipulations used in later Methods courses.
I feel even worse because I have at least 20 books on special functions which serve no purpose apart from looking good on my library self and dust them once a year.

On a serious note I started revisiting undergraduate stuff since last August in order to add resources for students on my website. It certainly feels good, as you mentioned, doing "hard maths" after such a long time.

I will add files/resources on special functions eventually, as I will have a lot of time to fill in.

Then I will match you.

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Updated: January 16, 2015
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