Burcu SilindirSeçil GergünAhmet YantirGergün, SeçilSilindir, BurcuYantir, Ahmet2025-10-06202419350090, 232503991935-00902325-039910.18576/amis/1805122-s2.0-85200407128https://www.scopus.com/inward/record.uri?eid=2-s2.0-85200407128&doi=10.18576%2Famis%2F180512&partnerID=40&md5=91bd5c7245af87753eecd55fb0c0d508https://gcris.yasar.edu.tr/handle/123456789/8284https://doi.org/10.18576/amis/180512This article is devoted to present the α-power function calculus on α-time scale the α-logarithm and their applications on α-difference equations. We introduce the α-power function as an absolutely convergent infinite product. We state that the α-power function verifies the fundamentals of α-time scale and adheres to both the additivity and the power rule for α-derivative. Next we propose an α-analogue of Cauchy-Euler equation whose coefficient functions are α-polynomials and then construct its solution in terms of α-power function. As illustration we present examples of the second order α-Cauchy-Euler equation. Consequently we construct α-analogue of logarithm function which is determined in terms of α-integral. Finally we propose a second order BVP for α-Cauchy-Euler equation with two point unmixed boundary conditions and compute its solution by the use of Green’s function. © 2024 Elsevier B.V. All rights reserved.Englishinfo:eu-repo/semantics/closedAccessBvp, Green’s Function, Α-cauchy-euler Equation, Α-logarithm, Α-power Function, Α-time Scale CalculusΑ-logarithmα-Power Functionα-Cauchy-Euler Equationα-Time Scale CalculusBVPGreen’s FunctionArticle