Jacobi-Anger expansion of Fourier series

Question

I know well that the Jacobi-Anger expansion is

$$e^{iz\sin (x)}=\sum_{n=-\infty}^\infty J_n(z)e^{inx}$$

or

$$e^{iz\cos (x)}=\sum_{n=-\infty}^\infty i^n J_n(z)e^{inx}$$

My question is how to expand with Jacobi-Anger expansion if the subscript of the exponential is a Fourier series? For example what is the solution if the subscript is the first two terms of the Fourier series of a triangular function:

$$e^{iz\cos (x)+iz\frac{1}{9}\cos(3x)}=\text{?}$$

Thanks!