This proposed project uses two classes of broad mathematical techniques, combinatorics and analysis, to expand knowledge of special functions in two ways. First it introduces new classes of special functions and catalogue their properties. Second, it studies new properties of known special functions. In particular, this project comprises ten subprojects. These include discovering and classifying algebraic relationships between special values of multidimensional polylogarithms, studying the combinatorics of shuffles and their q-analogues, analyzing the almost everywhere behavior of the Rogers-Ramanujan continued fraction on the unit circle, and studying continued fractions with multiple limit values.
Special functions are those mathematical functions which have recurred often enough that they have been distinguished with their own name and have had their properties catalogued. Indeed, the famous mathematician Paul Turan once said that special functions would be better named "useful functions". The most famous examples are perhaps the trigonometric functions, which are of critical use in all areas of technology. As time progresses and physical, technological, and mathematical knowledge grow, the set of functions dignified with the term "special" also gets larger. Additionally, new properties of functions already in the pantheon of special functions are discovered and these properties are added to the banks of human knowledge. This project investigates certain natural new special functions as well as explores novel properties of known special functions.
