Question:
I came across this sum:
$$f\left(\phi,x\right)\equiv \sum_{n=-\infty}^\infty e^{in\phi}J_n(x)$$ where $\phi$ and $x$ are real numbers. We know that $f(0,x)=\sum_{n=-\infty}^\infty J_n(x)=1$.
Because $\left|e^{in\phi}\right|\le 1$, I was hoping to be able to prove an upper-bound, something of the form:
$$\left|f\left(\phi,x\right)\right|\overset ?\le 1$$ or $$\left|f\left(\phi,x\right)\right|\overset ?\le g(\phi,x)$$ where $g(\phi,x)$ has a closed form.
Proving bounds for the odd and even sums would also be useful for me:
$$O\left(\phi,x\right)\equiv\sum_{n=-\infty}^\infty e^{i(2n-1)\phi}J_{2n-1}(x)\\ E\left(\phi,x\right)\equiv\sum_{n=-\infty}^\infty e^{i2n\phi}J_{2n}(x)\\ $$ $$\Rightarrow\left|O(0,x)\right|\overset ? \le1 \& \left|E(0,x)\right| \overset ? \le 1 $$
Any help is appreciated.
Some Background:
This series happens to arise when one tries to get a more accurate solution to the two-level harmonically driven system. Historically, people used to approximate the solution to these systems with the Rotating Wave Approximation. However, last year there was a paper giving a more accurate solution, specially for the strongly driven systems. These series arise when one tries to calculate the ground/excited state amplitudes of the system.
By the Jacobi-Anger expansion $$ e^{iz\cos\theta}=\sum_{n=-\infty}^{+\infty}i^n J_n(z) e^{in\theta}\tag{1} $$ hence by substituting $\theta=\frac{\pi}{2}-\alpha $ we get $$ e^{iz\sin\alpha} = \sum_{n=-\infty}^{+\infty} (-1)^n J_n(z) e^{-in\alpha}\tag{2} $$ and by substituing $\varphi=\pi-\alpha$ and $z=x$ we get $$ f(\varphi,x)=\sum_{n=-\infty}^{+\infty}J_n(x) e^{in\varphi} = \color{red}{\large e^{ix\sin\varphi}}\tag{3} $$ from which $\left|f(\varphi,x)\right|=1$ for any $x,\varphi\in\mathbb{R}$.