Does the Bessel Function Identity J_n-1(z) + J_n+1(z) = (2n/z) J_n(z) Hold?

[bdj03001]

Complex analysis: Let J_n (z) be the Bessel function for a positive integer n of order n. Verify?

J_n-1 (z) + J_n+1 (z) = ((2n)/z) J_n (z)

 

[Orion1]

Bessel Identity...


For a Bessel function of the first kind J_n(z)

Identity confirmed:
J_{n-1}(z) + J_{n+1}(z) = \frac{2\,n\,J_{n}(z)}{z} \; \; \; n > 0 \; \; \; z \neq 0

\Mfunction{BesselJ}(-1 + n,z) + \Mfunction{BesselJ}(1 + n,z) = \frac{2\,n\,\Mfunction{BesselJ}(n,z)}{z} \; \; \; n > 0 \; \; \; z \neq 0

n = 1
Attachment 1: LHS plot
Attachment 2: RHS plot

The x-intercepts and amplitudes appear to match, therefore this is an identity.
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Reference:
http://www.efunda.com/math/bessel/besselJYPlot.cfm