Is a formal power series a fraction of polynomials?

Question

We consider formal power series over a field. Power series such as $x+x^2+ x^3 +\cdots$ can be expressed as fraction of polynomials where the denominator is a unit in the power series ring. There are many other examples that we can construct. Does there exist a power series which cannot written as fraction of polynomials? Are there are some explicit examples?

Answer 1

You are asking for a formal power series which is not rational. In fact almost all formal power series are irrational, closely analogous to how almost all real numbers are irrational. There are many ways to write down counterexamples over different fields; here are a few.

Exercise 1: Over a finite field $\mathbb{F}_q$, any rational power series $\frac{f(x)}{g(x)} = \sum_{a \ge 0} a_n x^n$ has the property that $a_n$ must be eventually periodic.

This is a nice exercise and is the precise analogue of the fact that a rational real number must have an eventually periodic decimal expansion. It follows that any formal power series over $\mathbb{F}_q$ which is not eventually periodic is irrational; an easy example is $\sum_{n \ge 0} x^{n^2}$.

Exercise 2: If $F$ is a finite or countable field, then there are only countably many rational functions over $F$, but uncountably many formal power series over $F$.

This is not explicit but is one way to formalize the idea that "almost all" formal power series are irrational. For example this result applies to $F = \mathbb{Q}$.

Exercise 3: If $F$ is a subfield of $\mathbb{C}$, then any rational power series $\frac{f(x)}{g(x)} = \sum_{a \ge 0} a_n x^n$ has the property that $|a_n|$ grows at most exponentially.

It follows that any formal power series over $\mathbb{C}$ whose coefficients grow faster than exponentially cannot be rational, such as $\sum_{n \ge 0} n! x^n$. Much more precise versions of this claim can also be proven, e.g. $a_n$ must in fact grow exactly like an exponential times a polynomial so also cannot have any other growth rate (so e.g. $a_n$ can't grow like $\sqrt{n}$ or $\exp(\sqrt{n})$), and $a_n$ must in fact have a very specific closed form that most sequences don't have.

Exercise 4: If $F = \mathbb{Q}$, then any rational power series $\frac{f(x)}{g(x)} = \sum_{a \ge 0} a_n x^n$ has the property that at most finitely many primes divide the denominators of each $a_n \in \mathbb{Q}$.

It follows that any formal power series over $\mathbb{Q}$ such that the denominators are divisible by infinitely many primes is irrational, such as $\exp(x) = \sum_{n \ge 0} \frac{x^n}{n!}$ and $\ln (1 + x) = \sum_{n \ge 1} \frac{(-1)^{n-1} x^n}{n}$. Actually these guys are irrational over every field of characteristic $0$ but the proof is a bit trickier.

For the next exercise, say that a power series is lacunary if there are arbitrarily large gaps of zero coefficients between its nonzero coefficients; $\sum_{n \ge 0} x^{n^2}$ is a typical example.

Exercise 5: Every rational power series $\frac{f(x)}{g(x)} = \sum_{a \ge 0} a_n x^n$ has the property that if $\deg g$ consecutive coefficients are zero then it is a polynomial. Hence every lacunary power series is irrational.

This requires no hypotheses on the ground field so, for example, $\sum_{n \ge 0} x^{n^2}$ is irrational over every field.