>> bessel_j(nu-1,z)+bessel_j(nu+1,z) = (2*nu/z) * bessel_j(nu,z)
>...
>> After substituting a solution into a differential equation
>> I have the following expression that should equal zero....
>
>I don't know of any straightforward way to do this using patterns, or
>any other way to do it automatically (automagically as we say), but
>here's the approach I would take. For simplicity, I will use b(n) as a
>shorthand for bessel_j(n,x^2/2). By the way, in my version of Maxima
> (5.9.0), the arguments to bessel_j are in the other order! anyway....
Thanks for the ideas. I will investigate further.