This research delves into the practical uses of fractional calculus and generalised hypergeometric functions, illuminating how these tools may be used to expand traditional mathematical ideas and tackle difficult analytical issues. For a long time, hypergeometric functions were fundamental in both theoretical and practical mathematics because of the large class of special functions they could express. This paper proves the integral, differential, Mellin, and Laplace transformations of these functions and extends them to hyperbolic forms. We construct novel representations of hyperbolic functions, which are more general and useful, by using extended beta and gamma functions. To prove theorems rigorously and develop corollaries that emphasise the features of the extended functions, the technique integrates advanced mathematical methods. Modern applications in physics, engineering, and applied sciences are enriched by the study, which proves the usefulness of fractional calculus in expanding discoveries that were previously limited to integer-order scenarios. The research makes substantial contributions to mathematical analysis and practical problem-solving by bridging the gap between abstract theory and real-world situations using this technique. Extending fractional calculus to nonlocal, memory-based, generalised systems and generalised hypergeometric functions as a versatile analytical framework are both emphasised in the paper.