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arXiv · 2202.10782

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

Makoto Kawashima·Anthony Poëls
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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.

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Makoto Kawashima, Anthony Poëls. 2022-02-22. Pad\'e approximation for a class of hypergeometric functions and parametric geometry of numbers. https://doi.org/10.1016/j.jnt.2022.05.009

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