Hypergeometric Functions, How Special Are They?

where a, b, c are rational parameters. By specialization 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 taking a = b = 1/2, c = 3/2 gives us arcsin(√z)/√z. Finally, taking all parameters equal to 1 recovers the ordinary geometric series, which more or less explains the name hypergeometric series that was given by Euler to his series. Hypergeometric 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