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Topic: Gamma Function - Bessel Function Identity
Replies: 3   Last Post: Sep 24, 2012 5:30 AM

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Herman Rubin

Posts: 309
Registered: 2/4/10
Re: Gamma Function - Bessel Function Identity
Posted: Sep 7, 2012 4:35 PM
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On 2012-09-06, ksoileau <kmsoileau@gmail.com> wrote:
> I have observed and proved the following identity for all x\ne 0 :
> $$
> (I_{k-1}(x) +I_{k+1}(x) )I_{-k}(x)
> -(I_{-k-1}(x)+I_{-k+1}(x))I_k(x)
> = \frac{4 k}{x \Gamma (1-k) \Gamma (1+k)}
> $$
>
> Is this well-known or trivially derived? Any comments will be appreciated.
> Thanks,
> Kerry M. Soileau


I am unaware of it, but the right side can be greatly
simplified to 4 sin(k\pi)/(x\pi); it might be known in this form.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558




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