Theory of Hypergeometric Functions pp 1-19 | Cite as

# Introduction: the Euler−Gauss Hypergeometric Function

## Abstract

The binomial series \({(1 + x)}^{\alpha } ={ \sum \nolimits }_{n=0}^{\infty }\frac{\alpha (\alpha - 1)\cdots (\alpha - n + 1)} {n!} {x}^{n},\quad \vert x\vert < 1\) is the generating function of binomial coefficients \(\left (\begin{array}{*{10}c} \alpha \\ n \end{array} \right )=\frac{\alpha (\alpha - 1)\cdots (\alpha - n + 1)} {n!}.\)A hypergeometric function can be regarded as a generating analytic function of more complicated combinatorial numbers which generalizes the binomial series. By studying its analytic structure, it provides us with information such as relations among combinatorial numbers and their growth. The aim of this book is to treat hypergeometric functions of several variables as complex analytic functions. Hence, we assume that the reader is familiar with basic facts about complex functions.