Date: Sep 6, 2012 10:43 AM
Author: ksoileau
Subject: Gamma Function - Bessel Function Identity

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