Discrete analogs of the Lebedev–Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels $\mathrm{R}\mathrm{e}{K}_{\alpha +in}\left(x\right),\mathrm{I}\mathrm{m}{K}_{\alpha +in}\left(x\right),x>0,n\in \mathbb{N},|\alpha |<1,i$ is the imaginary unit and $K_{\nu }\left(z\right)$ 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.

中文翻译：

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离散 Lebedev-Skalskaya 变换

介绍并研究了 Lebedev-Skalskaya 变换的离散类似物。它涉及关于核的级数和积分$\mathrm{电阻}\mathrm{电子}{钾}_{\alpha +\mathrm{一世}n}\left(X\right),\mathrm{一世}\mathrm{米}{钾}_{\alpha +\mathrm{一世}n}\left(X\right),X>0,n\in \mathbb{N},|\alpha |<1,\mathrm{一世}$ 是虚数单位和 ${钾}_{\nu }\left(z\right)$是修正的贝塞尔函数。根据这些级数和积分，建立合适的函数和数列的相应反演公式，当$\alpha =\pm 1/2$. 案子$\alpha =0$ 简化为 Kontorovich-Lebedev 变换。