The problem is to prove the following:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\sum_{m>0}J_{j+m}(x)J_{j+m+n}(x) = \frac{x}{2n}\left(J_{j+1}(x)J_{j+n}(x) - J_{j}(x)J_{j+n+1}(x)\right).[/tex]

Now for the rambling...

I've been reading for a while, but this is my first post. Did a quick search, but I didn't find anything relevant. I could have missed it, though.

I was reading a paper on quantum spin chains (Antal et al, Phys. Rev E, 59, 4912 (1999)) and a series of Bessel functions suddenly turned into a compact expression (shown above) without any mention of what was done. If you have access to the journal, this appears in the appendix on the last page... eq. (B2).

As a good chunk of the paper dealt with the details of evaluating the asymptotics of another series, it makes me think I really missed something simple since they just inserted this result without even a reference to Abramowitz/Stegun or Gradshteyn. After a month or so of being involved with this stuff, I came up with an extremely cumbersome "proof." I am by no means an expert on Bessel functions, so I just thought I'd ask if anyone who is more familiar with these manipulations knows a slick way to sum that series that doesn't take 2 pages. As an interesting corollary, if you differentiate the series, use some recurrence relations, and then equate it to the given result, you wind up with

[tex]\frac{1}{2}\int_{0}^{x}\left(J_{j+1}(x')J_{j+n}(x')+J_{j}(x')J_{j+n+1}(x')\right)dx' = \frac{x}{2n}\left(J_{j+1}(x)J_{j+n}(x)-J_{j}(x)J_{j+n+1}(x)\right).[/tex]

I haven't been able to find a direct proof for this either... don't know which part is easier to attack, but I thought it was worth a mention.

Thanks for reading.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Summation of a series of bessel functions

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**