Infinite sum of Spherical Bessel function and Cosine at odd index

Question

I've recently encountered the expression \begin{equation} \sum_{n = 0}^\infty J_{2n+1}(a)\cos[(2n+1)b]. \end{equation}

I'm famililar with the even form of this expression, which as a closed form solution $$J_0(a) + \sum_{n = 1}^\infty J_{2n}(a)\cos(2nb) = \cos(a\sin(b))$$

and was wondering if anyone had any clever ideas on how to obtain a closed form solution to its odd counterpart.

Answer 1

You are looking for the Jacobi–Anger expansion.

$$\Sigma_{n = 0}^\infty (-1)^{n}J_{2n+1}(a)\hspace{0.3mm}\text{cos}[(2n+1)b] = \frac{1}{2}\sin(a\cos(b))$$

Answer 2

Looks like dosen't exist a closed form.

I found this example on book page: 664 example: 6.

I have another formulas:

$$\color{blue}{\sum _{n=0}^{\infty } J_{2 n+1}(a) \cos ((2 n+1) b)}=\\\mathcal{L}_s^{-1}\left[\frac{s \cos (b)}{\sqrt{1+s^2} \left(1+2 s^2-\cos (2 b)\right)}\right](a)=\color{blue}{\\\frac{1}{2} \cos (b) \int_0^a \cosh \left(\frac{1}{2} (a-t) \sqrt{-2+2 \cos (2 b)}\right) J_0(t) \, dt}$$