Hi This is one of the problems for my take home final exam on differential equations.(adsbygoogle = window.adsbygoogle || []).push({});

I have been looking for a solution for this problem intensely for the last two days. This problem comes from Calculus vol 2 by Apostol section 6.24 ex 7. here it is

1. The problem statement, all variables and given/known data

Use the identities in exercise 6 to show that:

[tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}=1[/tex]

and

[tex]\sum^{\infty}_{n=0}(2n+1)J^{2}_{n}J^{2}_{n+1}=\frac{1}{2}x[/tex]

2. Relevant equations

The identities that I already proved in exercise 6 are:

[tex]\frac{1}{2}\frac{d}{dx}(J^{2}_{\alpha}+J^{2}_{\alpha+1})=\frac{\alpha}{x}J^{2}_{\alpha}-\frac{\alpha+1}{x}J^{2}_{\alpha+1}[/tex]

and

[tex]\frac{d}{dx}(xJ^{2}_{\alpha}J^{2}_{\alpha+1})=x(J^{2}_{\alpha}-J^{2}_{\alpha+1})[/tex]

3. The attempt at a solution

As I mentioned in the introduction I have looked for possible solutions and background information for a while but I'm still stuck.

The best source for information on this problem is http://books.google.com/books?id=d4...SQ&sig=t6mTG7P19IxaJ797Ri-1NojAgWc#PPA361,M1" but I also don't undertstand how hansen changes the from

[tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}[/tex]

to

[tex]\left(e^{\frac{1}{2}z\frac{t-1}{t}}\right)\left(e^{\frac{1}{2}z\frac{-t+1}{t}}\right)[/tex]

I think the problem is that I just don't have enough knowledge of power series to know how to handle this problem. I just don't know how to approach the problem. I though t that maybe:

[tex]2\sum^{\infty}_{n=1}\frac{J^{2}_{0}}{2}+J^{2}_{n}[/tex](is this even right?)

and then try to change it into something that looks like a recurrence relation

other approach I tried was to expand the series and try recurrence relations there. but I got stuck there too.

Thanks very much for any help or links to relevant information.

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# Homework Help: Prove a sum identity for bessel function

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