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