Integral representations for Neumann-type series of Bessel functions \(I_{\nu},Y_{\nu}\) and \(K_{\nu}\) (Q2880656)

The authors establish integral representations for Neumann-type series of modified Bessel functions of the first and second kind \(I_{\nu}\) and \(K_{\nu}\), Bessel functions of second kind \(Y_{\nu}\) and Hankel functions \(H_{\nu}^{(1)}, H_{\nu}^{(2)}\). They present closed form expressions for these Neumann series. Their tools are the Laplace integral form of a Dirichlet series, the condensed form of the Euler-Maclaurin summation formula and certain bounding inequalities for \(I_{\nu}\) and \(K_{\nu}\). The analogous results for Bessel functions of first kind \(J_{\nu}\) has been proved by \textit{T. K. Pogány} and \textit{E. Süli} [Proc. Am. Math. Soc. 137, No. 7, 2363--2368 (2009; Zbl 1171.33003)].