Exploring generalized discrete Bessel functions and cylindrical Laplace equation on α-time scale

This article introduces a novel approach for discrete Bessel equations and Bessel functions of the first and second kinds on α-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 α-Bessel equation of order [ν]α, construct α-Bessel function of order [ν]α of the first kind, Jνα, and analyze second linearly independent solutions in two distinct cases. We introduce α-Bessel function of order [ν]α of the second kind, namely α-Weber's Bessel function, Yνα, as a special linear combination of Jνα and Ȳνα, where Ȳνα is α-Neumann's Bessel function. The α-Bessel function of the second kind recovers the various Bessel functions of the second kind in proper limits. Additionally, we present modified α-Bessel equation and functions of the first and second kinds and state a relationship between first kinds of α-Bessel and modified α-Bessel functions. As an application, we introduce the α-analogue of the Laplace equation in cylindrical coordinates and derive its solutions in terms of α-analogues of trigonometric, hyperbolic, Bessel, and modified Bessel functions. Finally, graphs of some α-Bessel functions are presented and reductions to several discrete Bessel equations and functions are outlined.