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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?

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The $J$ Bessel function does, in fact, satisfy a multitude of functional equations, many of them involving Riemann Sums. For example, one such functional equation is $$J_m(x+y)=\sum_{n=-\infty}^\infty J_n(x)J_{m-n}(y)$$ Which is known as the "Bessel Function Addition Theorem". It also satisfies the functional equations $$\sum_{n=-\infty}^\infty J_n(x)=1$$ and $$J_0(x)^2+2\sum_{n=0}^\infty J_n(x)^2=1$$

There are many other such identities, and you can find them at this page.

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