Some Model Theory of Hypergeometric and Pfaffian Functions

Abstract

We present some results and open problems related to expansions of the field of real numbers by hypergeometric and related functions focussing on definability and model completeness questions. In particular, we prove the strong model completeness for expansions of the field of real numbers by the exponential, arctangent and hypergeometric functions. We pay special attention to the expansion of the real field by the real and imaginary parts of the hypergeometric function ${_2F_1}({1}/{2},{1}/{2};1;z)$ because of its close relation to modular functions.