Summing Sneddon-Bessel series explicitly
Antonio J. Durán, Mario Pérez, Juan L. Varona
TL;DR
The paper provides explicit closed-form expressions for Sneddon-Bessel series $S_q^{\alpha,\beta,\nu}(x,y)$ with $q_n=2n+\alpha+\beta-2\nu+2$, using a residue-based partial fraction framework for $\frac{f(z)}{\Phi_\nu(z)^2}$ and a generating-function decomposition into $\delta_n^{\alpha,\beta,\nu}$ and $\phi_n^{\alpha,\beta,\nu}$. It derives a PDE-driven approach that yields exact formulas for all $n\ge0$ (including $n=0,1$) and negative $n$, and extends these results to the one-variable case and to generalized Kneser-Sommerfeld expansions. The work unifies and extends classical summations of Bessel-series by expressing the sums in terms of Gamma functions, hypergeometric functions, and Bessel-type transforms, with rigorous parameter-domain conditions and analytic continuation. The resulting identities enable precise evaluation of Sneddon-Bessel sums and provide new extensions of the Kneser-Sommerfeld expansions, with potential applications in mathematical physics and engineering problems involving radially symmetric solutions and spectral decompositions.
Abstract
We sum in a close form the Sneddon-Bessel series \[ \sum_{m=1}^\infty \frac{J_α(x j_{m,ν})J_β(y j_{m,ν})} {j_{m,ν}^{2n+α+β-2ν+2} J_{ν+1}(j_{m,ν})^2}, \] where $0<x$, $0<y$, $x+y<2$, $n$ is an integer, $α,β,ν\in \mathbb{C}\setminus \{-1,-2,\dots \}$ with $2\operatorname{Re} ν< 2n+1 + \operatorname{Re} α+ \operatorname{Re} β$ and $\{j_{m,ν}\}_{m\geq 0}$ are the zeros of the Bessel function $J_ν$ of order $ν$. As an application we prove some extensions of the Kneser-Sommerfeld expansion.