# Why are these two series of Bessel functions equal?

I noticed that the following expression holds true for $$k \in \left[-1,1\right]$$:

$$\operatorname{J}_{0}\left(x\right) + 2\sum_{n = 1}^{\infty}{\rm i}^{n}\operatorname{J}_n\left(x\right)\cos\left(n \cos^{-1}\left(k\right)\right) = \operatorname{J}_{0}\left(kx\right) + 2\sum_{n = 1}^{\infty}{\rm i}^{n} \operatorname{J}_{n}\left(kx\right),$$ where $$\operatorname{J}_{n}$$ is the Bessel function of the first kind and $${\rm i} = \sqrt{-1}$$ is the imaginary unit.

It is related to the Jacobi-Anger expansion. I stumbled upon this equality, it fascinates me, and I don't understand how it is equal. I have checked numerically for many values of $$x$$.

Appreciate any advice or explanation $$!$$.

$$\cos(n\cos^{-1}(k))$$ is a Chebyshev polynomial evaluated at $$k$$.
It looks like one side uses a scaled version of the other. So the multiplication theorem may come into play: $$\begin{equation*} \lambda^{-\nu}J_\nu(\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{(1-\lambda)^2z}{2} \right)^n J_{\nu+n}(z) \end{equation*}$$