|
Hypergeometric Functions
A hypergeometric function is a power series
Σ cn zn
of one complex variable, in which the ratio of successive coefficients
cn+1 / cn
is a rational function of
n.
Classical hypergeometric
functions were studied by the likes of Euler, Gauss, Kummer and
Riemann, beginning in the early 18th century, and satisfy many
beautiful identities and transformation properties. They are
versatile functions with numerous applications. For instance,
depending on how we specialize their parameters, they may give
solutions to differential equations, expressions for Bessel functions
or orthogonal polynomials, or periods of elliptic curves.
Hypergeometric functions
defined over finite fields were introduced
in the 1980's, and have proven to be just as intriguing as their
classical cousins. Among other things, they have been used to count
points mod
p
on elliptic curves, and to give interesting
congruences for Ramanujan's tau-function. In this talk, we will
explore some of these fascinating properties, both for the classical
and finite-field hypergeometric functions.
|
