Sum involving exp of squared zeros of Bessel functions

Question

Let $m,n$ be integers, and $j_{n,m}$ be the $m$-th positive zero of the $n$-th Bessel function $J_n$. How do we compute the sum

$$ \sum_{m=1}^\infty \exp \left( - A \, j_{n,m}^2 \right)$$

for some $A >0$ ?

Answer 1

Here's a start.

The approximate zeros of $J_v$ follow from the asymptotic expansion

$J_v(z) =\sqrt{\frac{2}{\pi x}}\cos(z-\frac12 \pi v-\frac14 \pi) $.

This is zero when $z-\frac12 \pi v-\frac14 \pi =\pi n +\frac12 \pi $ or $z_{v, n} =\frac12 \pi v+\frac14 \pi +\pi n +\frac12 \pi =\pi(n+\frac12 v+\frac34 ) $.

Since this grows linearly with $n$, your $A$ must be negative.

This starts to look like a theta function which has a transformation formula which might enable the sum to be more conveniently evaluated.

The terms in the sum get rapidly smaller, so most of the value will be contributed by the first few terms. These, of course, are the ones where the asymptotic approximation is least accurate.