A determinantal approach to irrationality

Abstract

It is a classical fact that the irrationality of a number $\xi \in \mathbb{R}$ follows from the existence of a sequence $p_n/q_n$ with integral $p_n$ and $q_n$ such that $q_n\xi -p_n\ne 0$ for all $n$ and $q_n\xi -p_n\to 0$ as $n\to \infty$. In this note we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement $q_n\xi -p_n\to 0$ is weakened. Some applications are given including a new proof of the irrationality of $\pi$. Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.

9 pages

References in corpus (4)

• A few remarks on linear forms involving Catalan's constant
• On the non-quadraticity of values of the q-exponential function and related q-series
• Hankel determinants of Dirichlet series
• On the irrationality of generalized $q$-logarithm

Cited by in corpus (4)

• Euler's factorial series and global relations
• On the irrationality of generalized $q$-logarithm
• Vectors of type II Hermite-Padé approximations and a new linear independence criterion
• Rational Approximations via Hankel Determinants