Some Model Theory of Hypergeometric and Pfaffian Functions

TL;DR

This paper investigates the definability and model completeness of real number fields expanded by hypergeometric and related functions, proving strong model completeness for certain expansions including exponential, arctangent, and hypergeometric functions.

Contribution

It establishes strong model completeness for expansions of the real field by specific functions, including hypergeometric functions related to modular functions.

Findings

Proves strong model completeness for expansions with exponential, arctangent, and hypergeometric functions.

Analyzes the definability of hypergeometric functions in real fields.

Highlights the relation between hypergeometric functions and modular 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.