On the coefficients of Neumann series of Bessel functions (Q536265)

In this note the authors describe the class \(\Lambda=\{\alpha\}\) of functions that generate the integral representation of a Neumann series of Bessel functions, in the sense that the restriction \(\alpha|_{\mathbb N}=(\alpha_{n})\) forms the coefficient array of the series \[ {\mathcal N}_{\nu}(z)=\sum_{n=1}^{\infty}\alpha_{n}J_{\nu+n}(z), \quad z\in \mathbb C, \] where \(\nu\), \(\alpha_n\) are constants and \(J_\mu\) is the Bessel function of the first kind and order \(\mu\). Knowing only the set of nodes \(N:=\{(n,\alpha_n)\}_{n\in\mathbb N}\), this question cannot be answered merely by examining the convergence of the series \({\mathcal N}_\nu(z)\) and then interpolating the set \(N\). The authors formulate an answer to this question so that the resulting class of functions \(\alpha\) depends on a suitable, integrable on \(\mathbb R_+\), scaling-function \(h\).