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

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Tom D

Posts: 7
Registered: 8/4/10
Re: Gamma Function - Bessel Function Identity
Posted: Sep 24, 2012 5:30 AM
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It seems to me you could derive this by using the Wronskian for Bessel
functions (+ some massaging) towork out the LHS, which will come out
with a sin() function in it, then you could rewrite the sin() as two
Gamma functions by using the reflection formula for the gamma
function, as in:

Gamma(1+k) Gamma(1-k) = k Gamma(k) Gamma(1-k) = k pi/sin(pi k).

Hope this is of some help,
Tom




On Thu, 6 Sep 2012 14:43:32 +0000, 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





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