Pad\'e approximation for a class of hypergeometric functions and parametric geometry of numbers

TL;DR

This paper develops new methods using Padé approximations and parametric geometry of numbers to derive effective irrationality measures for certain hypergeometric function values, with potential arithmetic applications.

Contribution

It introduces explicit Padé approximations for hypergeometric functions and establishes an effective Poincaré-Perron theorem to analyze their asymptotics.

Findings

Derived effective irrationality measures for binomial function values at rational points

Constructed Padé approximations satisfying Poincaré-type recurrence relations

Proved a general theorem on simultaneous rational approximations using parametric geometry of numbers

Abstract

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We explicitly construct Pad\'e approximations by using a formal method and show that the associated sequences satisfy a Poincar\'e-type recurrence. To study precisely the asymptotic behavior of those sequences, we establish an \emph{effective} version of the Poincar\'e-Perron theorem. As a consequence we obtain, among others, effective irrationality measures for values of binomial functions at rational numbers, which might have useful arithmetic applications. A general theorem on simultaneous rational approximations that we need is proven by using new arguments relying on parametric geometry of numbers.