Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$
And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$.
At the same time, there is no known functional equation for Bessel J $J(z)$ function. Or at least, I don't know such an equation.

Is there some kind of relation between the function, differential equations and functional equations? Is it possible to prove that every function is a solution of some functional equation? Is it possible to construct the equation from known differential equation or function definition through series?