Hypergeometric series refer to power series whose ratio of successive coefficients are a rational function of the series' index variable. The "2F1" hypergeometric series was famously treated by Gauss, and interest in them has survived to modern times.
Hypergeometric series can be thought of as a "library" of the most common functions in mathematics. For example, the "0F0" function is an exponential, the "2F1" can be both the logarithm and arcsin, and the "0F1" can be sine or cosine but also Bessel functions. Numerous sequences of orthogonal polynomials also have representations as hypergeometric functions.
This talk will begin with exposition on the classical hypergeometric series "pFq" and some of its special properties. Following this, there will be a discussion on discrete analogues of these functions invented by the speaker and some of their applications to the theory of difference equations as well as their relation to classical hypergeometric series.