Exploring generalized discrete Bessel functions and cylindrical Laplace equation on $\alpha$-time scale

This article introduces a novel approach for discrete Bessel equations and Bessel functions of the first and second kinds on $\alpha$-time scale. This approach not only addresses existing gaps in discrete, $q$-discrete and $(q,h)$-discrete Bessel equations and functions of the second kind but also provides their unification and extension. We introduce $\alpha$-Bessel equation of order $[\nu]\alpha$, construct $\alpha$-Bessel function of order $[\nu]\alpha$ of the first kind, $J_{\nu}^{\alpha}$, and analyze second linearly independent solutions in two distinct cases. We introduce $\alpha$-Bessel function of order $[\nu]\alpha$ of the second kind, namely $\alpha$-Weber's Bessel function, $Y{\nu}^{\alpha}$, as a special linear combination of $J_{\nu}^{\alpha}$ and $\bar{Y}{\nu}^{\alpha}$, where $\bar{Y}{\nu}^{\alpha}$ is $\alpha$-Neumann's Bessel function. The $\alpha$-Bessel function of the second kind recovers the various Bessel functions of the second kind in proper limits. Additionally, we present modified $\alpha$-Bessel equation and functions of the first and second kinds and state a relationship between first kinds of $\alpha$-Bessel and modified $\alpha$-Bessel functions. As an application, we introduce the $\alpha$-analogue of the Laplace equation in cylindrical coordinates and derive its solutions in terms of $\alpha$-analogues of trigonometric, hyperbolic, Bessel, and modified Bessel functions. Finally, graphs of some $\alpha$-Bessel functions are presented and reductions to several discrete Bessel equations and functions are outlined.