This project focuses on mathematical research in the area of special functions and their applications to a multiplicity of scientific questions. Among the most important special functions which arise in applications fall within the general class of hypergeometric functions, which, for example, have recently found their way into the theory of quantum groups. In addition to linear and linear fractional transformations, some of these functions have quadratic or higher transformations. Quadratic transformations have been studied for two centuries and are reasonably well understood. Cubic transformations were first discovered by Riemann over 125 years ago, yet they are not as well understood as comments in the literature lead one to think, and while applications of them have been limited, there are some recent very interesting ones which go back to work of Ramanujan. These transformations will be analyzed. Efforts will be made to find multivariable extensions. Special functions are fundamental building blocks used in the approximation of complex functions, as solutions of differential equations and in the representation of many important mathematical and physical relationships. In addition to their recurrence in often unexpected ways throughout science, they generally are easy to compute or approximate numerically.
