A new approach to hypergeometric functions via $\alpha$-time scale

This manuscript investigates hypergeometric functions on a general discrete time scale $\mathbb{T}_\alpha$. To this end, we introduce the $\alpha$-beta function, establish its fundamental properties, and explore its connection with the $\alpha$-gamma function. In order to achieve our main objective, we formulate the $\alpha$-analogue of the hypergeometric equation and construct its solutions, referred as the $\alpha$-hypergeometric functions. In parallel with the classical case, we illustrate that the $\alpha$-hypergeometric function encompasses many $\alpha$-elementary functions. Furthermore, we derive several key properties of the $\alpha$-hypergeometric function, including the $\alpha$-analogues of the Gauss summation formula, Pfaff transformation, and Euler’s linear identity, which reduce to their classical and $q$-analogous forms.