Summing Sneddon-Bessel series explicitly

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