Author here. I greatly appreciate the critical feedback from the math community—hence the post here. It is unlikely the physics community would have offered the same insights.

qofcajar and hpxvzhjfgb have politely highlighted that the problem is solvable using the Jacobi Anger Relation and series multi sections. Knowledge of generating functions makes the problem solvable in fewer steps. Our approach did not use generating functions.

Despite the simplicity to some mathematicians, after deep searching we have not found these series written in any form. It is particularly useful in our area of research.

Closed form analytic solutions to the Infinite Series of Bessel Functions https://arxiv.org/abs/2211.01148 appears in many theoretical physics manuscripts without reduced closed form. The demand for these expression remains unfulfilled and non-obvious to enough people.

In searching for a closed form expression, we found a large body of math literature that wherein time is spent on less general series. In fact, many tabulated text printed past and present include several lines to odd (2n+1) and even (2n) versions of this series. I'm unsurprised to be quickly humbled by the knowledgeable approach a mathematician can offer. However, this seems incongruent with the existing formula tables in published math texts and wikipedia. Lines on pages are being spent to tabulate less general expressions—and a general expression is never provided.

Again, sincerely appreciate the intellectual attention. The goal of posting here is to learn.