Hypergeometric Functions, How Special Are They?

Abstract

Section 1. Introduction In the world of standard functions, the hyper-geometric functions take a prominent position in mathematics, both pure and applied, and in many branches of science. They were introduced by Euler as power series expansions of the form 1 + a · b c · 1 z + a(a + 1)b(b + 1) c(c + 1) · 1 · 2 z 2 + · · · , where a, b, c are rational parameters. By special-ization of the parameters, Euler obtained the various classical functions that were around at that time. For example, taking b = c = 1 gives us Newton's binomial series for (1 − z) −a and tak-ing a = b = 1/2, c = 3/2 gives us arcsin(√ z)/ √ z. Finally, taking all parameters equal to 1 recov-ers the ordinary geometric series, which more or less explains the name hypergeometric series that was given by Euler to his series. Hypergeo-metric functions also include functions that were entirely new in Euler's time. For example, taking a = b = 1/2, c = 1, one obtains the function 2 π 1 0 dx (1 − x 2)(1 − zx 2) , a so-called elliptic integral of the first kind. It is a period of the family of elliptic curves y 2 = (1 − x 2)(1 − zx 2) parameterized by z and is one of the most often quoted functions in algebraic geometry. Euler also found the hypergeometric equation, which is the second-order linear differ-ential equation that is satisfied by hypergeometric series. It reads z(z − 1)f + ((a + b + 1)z − c)f + abf = 0.