Discrete Lebedev-Skalskaya transforms

TL;DR

This paper introduces and analyzes discrete versions of the Lebedev-Skalskaya transforms, providing inversion formulas and connecting to the Kontorovich-Lebedev transform for specific parameter values.

Contribution

It develops discrete analogs of Lebedev-Skalskaya transforms and establishes their inversion formulas for certain parameter cases.

Findings

Derived inversion formulas for the discrete transforms.

Connected the case b1 1/2 to existing transforms.

Extended the framework to include the Kontorovich-Lebedev transform.

Abstract

Discrete analogs of the Lebedev-Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels ${\rm Re} K_{\alpha+in}(x), {\rm Im} K_{\alpha+in}(x), x >0, n \in \mathbb{N}, |\alpha | < 1,\ i $ is the imaginary unit and $K_\nu(z)$ is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established when $\alpha = \pm 1/2$. The case $\alpha=0$ reduces to the Kontorovich-Lebedev transform.