On approximation measures of $q$-exponential function

TL;DR

This paper develops effective approximation measures for $q$-exponential functions, providing explicit irrationality measures at rational points and improving bounds for specific rational approximations.

Contribution

It introduces new approximation measures for $q$-exponential functions and refines irrationality exponents for particular rational approximations.

Findings

Proves explicit irrationality measure for $q$-exponential at rational points.

Replaces Bundschuh's irrationality exponent 7/3 with approximately 2.1547 for certain rational approximations.

Provides effective bounds for approximations related to $q$-exponential functions.

Abstract

We shall present effective approximations measures for certain infinite products related to $q$-exponential function. There are two main targets. First we shall prove an explicit irrationality measure result for the values of $q$-exponential function at rational points. Then, if we restrict the approximations to rational numbers of the shape $d^s/N$, we may replace Bundschuh's irrationality exponent $7/3$ by $2+\frac{1}{3+2\sqrt 3}=2.1547...$.