Cubic theta functions and identities for Appell's F1 function.

摘要

This thesis is centered around three topics: the theory of the cubic theta functions as functions of two analytic variables, cubic modular equations, and a class of two-variable cubic modular equations. Chapter 2 is dedicated to the rst two topics, while Chapter 3 covers the last. First, the theory of cubic theta functions can be developed analogously to, but distinct from, the classical theory of elliptic functions. We will derive analogues of the Jacobian elliptic functions, and provide addition theorems, integral inversion formulas, di erential equations, and modular transformations for these functions. Second, we revisit the cubic modular equations rst derived by Ramanujan and study them in a systematic manner. The results obtained greatly extend previous work on cubic modular equations. Finally, in Chapter 3, we study modular equations for the Picard modular functions. These modular equations provide a two-variable generalization of the cubic modular equations studied in Chapter 2.