  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.
