Study of generalised hypergeometric functions and fractional calculus with applications

A strong foundation for tackling complicated issues in engineering, physics, and applied sciences is offered by the study of generalised hypergeometric functions and fractional calculus, which is a major junction of classical analysis and current mathematical theory. The generalised hypergeometric functions are naturally generated in the solutions of diverse differential equations and have rich analytical features. They expand the classic hypergeometric functions over a bigger parameter space. For memory-dependent and hereditary phenomena that cannot be effectively represented by standard integer-order calculus, fractional calculus offers more sophisticated tools by dealing with integrals and derivatives of arbitrary (non-integer) order. An examination of the foundations and relationships between fractional calculus and generalised hypergeometric functions, as well as their operational methods, integral transforms, and special function representations, is the goal of this research. An analysis of fractional differential equations analytically solved by generalised hypergeometric functions is highlighted, illustrating their practical use in problems of viscoelasticity, anomalous diffusion, and signal processing, among others. By providing concrete examples, we go deeper into the applications and show how these mathematical tools have contributed to theoretical advancements as well as practical applications. The importance of fractional-order operators and special functions in modern scientific research and mathematical modelling is highlighted by this study, which ultimately adds to our knowledge of their synergy.