Riordan Arrays, Elliptic Functions and their Applications

Riordan arrays have been used as a powerful tool for solving applied algebraic and enumerative combinatorial problems from a number of different settings in pure and applied mathematics. This thesis establishes relationships between elliptic functions and Riordan arrays leading to new classes of Riordan arrays which here are called elliptic Riordan arrays. These elliptic Riordan arrays were found in many cases to be useful constructs in generating combinatorially and algebraically significant sequences based on their corresponding trigonometric and hyperbolic forms. In addition, in some cases the elliptic Riordan arrays presented interesting structural patterns that were further investigated. By exploring elliptic Riordan arrays more closely with respect to other fields, several new applications of Riordan arrays associated with physics and engineering are illustrated. Furthermore, other non-elliptic type Riordan arrays having important applications are also presented based on the connection established in the thesis between Riordan arrays and the analytic solutions to some of the families of the Sturm-Liouville differential equations.