Summing Sneddon-Bessel series explicitly

TL;DR

This paper derives a closed-form sum for a specific Sneddon-Bessel series involving Bessel function zeros and applies the result to extend the Kneser-Sommerfeld expansion.

Contribution

It provides an explicit summation formula for the Sneddon-Bessel series and extends the Kneser-Sommerfeld expansion using this new result.

Findings

Closed-form expression for the Sneddon-Bessel series.

Extensions of the Kneser-Sommerfeld expansion.

Potential applications in mathematical analysis and physics.

Abstract

We sum in a close form the Sneddon-Bessel series \[ \sum_{m=1}^\infty \frac{J_\alpha(x j_{m,\nu})J_\beta(y j_{m,\nu})} {j_{m,\nu}^{2n+\alpha+\beta-2\nu+2} J_{\nu+1}(j_{m,\nu})^2}, \] where $0<x$, $0<y$, $x+y<2$, $n$ is an integer, $\alpha,\beta,\nu\in \mathbb{C}\setminus \{-1,-2,\dots \}$ with $2\operatorname{Re} \nu < 2n+1 + \operatorname{Re} \alpha + \operatorname{Re} \beta$ and $\{j_{m,\nu}\}_{m\geq 0}$ are the zeros of the Bessel function $J_\nu$ of order $\nu$. As an application we prove some extensions of the Kneser-Sommerfeld expansion.