Introduction

Abstract

The q-extension is the procedure to replace mathematical symbols, identities, functions and others by their meaningful q-analogues, and has been extensively studied. As a typical example of q-extension, here we introduce the q-Pochhammer symbol as the q-analogue of the well-known Pochhammer symbol. The main topic of the present monograph is not the q-extension, but its further extension; the elliptic extension. We introduce a parameter $$p \in \mathbb {C}$$ , $$0< |p| < 1$$ instead of q, which is called a nome, and then define the theta function parameterized by p, $$\theta (z; p)$$ , on the complex plane punctured at the origin, $$z \in \mathbb {C}^{\times }$$ . Basic properties of the theta function used throughout this monograph are summarized here.