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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


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(1k) = k Gamma(k) Gamma(1k) = 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 : >$$ > (I_{k1}(x) +I_{k+1}(x) )I_{k}(x) > (I_{k1}(x)+I_{k+1}(x))I_k(x) > = \frac{4 k}{x \Gamma (1k) \Gamma (1+k)} >$$ > >Is this wellknown or trivially derived? Any comments will be appreciated. >Thanks, >Kerry M. Soileau



