Pad? approximation for a class of hypergeometric functions and parametric geometry of numbers

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 Pade approximations by using a formal method and show that the associated sequences satisfy a Poincare-type recurrence. To study precisely the asymptotic behavior of those sequences, we establish an effective version of the Poincare-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.(c) 2022 Elsevier Inc. All rights reserved.