Gauss’ hypergeometric function

We give a basic introduction to the properties of Gauss’ hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hyperegeometric equation. Initially this document started as an informal introduction to Gauss’ hypergeometric functions for those who want to have a quick idea of some main facts on hypergeometric functions. It is the startig of a book I intend to write on 1-variable hypergeometric functions. As time progressed this informal note attracted increasing attention. Therefore I would like to add a point of WARNING here: right now the manuscript needs to be double-checked on possible errors again. At the moment I do not want to consider it as a solid reference. With this provision in mind you are welcome to read it (and let me know if you find errors) 1 Definition, first properties Let a, b, c ∈ R and c ̸∈ Z≤0. Define Gauss’ hypergeometric function by F (a, b, c|z) = ∑ (a)n(b)n (c)nn! z. (1) The Pochhammer symbol (x)n is defined by (x)0 = 1 and (x)n = x(x+ 1) · · · (x+ n− 1). The radius of convergence of (1) is 1 unless a or b is a non-positive integer, in which cases we have a polynomial. Examples. (1− z)−a = F (a, 1, 1|z) log 1 + z 1− z = 2zF (1/2, 1, 3/2|z) arcsin z = zF (1/2, 1/2, 3/2|z) K(z) = π 2 F (1/2, 1/2, 1, z) Pn(z) = 2 F (−n, n+ 1, 1|(1 + z)/2) Tn(z) = (−1)F (−n, n, 1/2|(1 + z)/2) Here K(z) is the Jacobi’s elliptic integral of the first kind given by