# Discrete Kontorovich-Lebedev transforms

## TL;DR

This paper introduces and studies discrete versions of the Kontorovich-Lebedev transforms involving Bessel functions, providing new expansions and solving a boundary value problem for the Helmholtz equation.

## Contribution

It presents the first discrete analogs of the classical Kontorovich-Lebedev transforms and applies them to solve a boundary value problem in the upper half-plane.

## Key findings

- Established new expansions of functions using the discrete transforms.
- Derived solutions to a Dirichlet boundary value problem for the Helmholtz equation.
- Analyzed properties of the discrete transforms involving Bessel functions.

## Abstract

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.