Hi everybody. I'm looking for an analogue of irrationality measure for formal power series with integer coefficient, the elements of $\mathbb{Z}[[x]]$. For any $f \in \mathbb{Z}[[x]]$ and positive integer $g$, I thought to define something like $$m_g(f) := \sup_{p,q} \mbox{ord}(f  p / q)$$ where $p,q \in \mathbb{Z}[x]$ satisfy $\deg p, \deg q \leq g$ and $q \neq 0$; $\mbox{ord}(h) := n_0$ for any formal Laurent series $h = \sum_{n=n_0}^\infty a_n x^n$ with $a_{n_0} \neq 0$ and $\mbox{ord}(0) := +\infty$. Note that $m_g(f)$ is finite for all $g$ if and only if $f$ is irrational, otherwise $m_g(f) = +\infty$ for $g$ sufficently large. Do you have any references on this? Thank you.
I think you will enjoy the paper "Irrationality of Power Series for Various Number Theoretic Functions", by W.D. Banks, F. Luca and I.E. Shparlinski. They use your $m_g$ as a measure of irrationality and give asymptotics on $m_g(f)$ for a variety of different power series $f$. They focus on power series with coefficients coming from arithmetic functions such as the Euler totient function, number of (prime, squarefree...) divisors, sum of divisors, Liouville function etc. 


Going by what's in Math Reviews, these two papers might be relevant: Sandra Delaunay, Approximation diophantienne et distances ultramétriques non standard, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 629–661, MR2188586 (2006i:11085). Dinesh S. Thakur, Diophantine approximation in finite characteristic, in Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 757–765, Springer, Berlin, 2004, MR2037123 (2004m:11109). 


Note that thanks to Pade approximation we have $m_g(f) \geq 2g + 1$ for all $g \geq 1$ and $f \in \mathbb{Z}[[x]]$. 

