Discrete analogs of the classical Kontorovich–Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function \(K_{in}(x), x >0, n \in {\mathbb {N}}, i \) is the imaginary unit, and incomplete Bessel functions. Several expansions of suitable functions and sequences in terms of these series and integrals are established. As an application, a Dirichlet boundary value problem in the upper half-plane for inhomogeneous Helmholtz equation is solved.

中文翻译：

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离散Kontorovich–Lebedev变换

介绍并研究了经典Kontorovich-Lebedev变换的离散类似物。它涉及具有修改后的Bessel函数或Macdonald函数\（K_ {in}（x），x> 0，n \ in {\ mathbb {N}}，i \）的级数，并且是不完整的Bessel函数。建立了关于这些系列和积分的适当功能和顺序的几种扩展。作为一个应用，解决了非均质亥姆霍兹方程的上半平面的狄利克雷边值问题。