Hi, I’m Maria and I’m a $q$-analog addict. The theory of $q$-analogs is a little-known gem, and in this series of posts I’ll explain why they’re so awesome and addictive!

So what is a $q$-analog? It is one of those rare mathematical terms whose definition doesn’t really capture what it is about, but let’s start with the definition anyway:

Definition: A $q$-analog of a statement or expression $P$ is a statement or expression $P_q$, depending on $q$, such that setting $q=1$ in $P_q$ results in $P$.

So, for instance, $2q+3q^2$ is a $q$-analog of $5$, because if we plug in $q=1$ we get $5$.

Sometimes, if $P_q$ is not defined at $q=1$, we also say it’s a $q$-analog if $P$ can be recovered by taking the limit as $q$ approaches $1$. For instance, the expression $$\frac{q^5-1}{q-1}$$ is another $q$-analog of $5$ – even though we get division by zero if we plug in $q=1$, we do have a well defined limit that we can calculate, for instance using L’Hospital’s Rule: $$\lim_{q\to 1} \frac{q^5-1}{q-1}=\lim_{q\to 1} \frac{5q^4}{1} = 5.$$

Now of course, there are an unlimited supply of $q$-analogs of the number $5$, but certain $q$-analogs are more important than others. When mathematicians talk about $q$-analogs, they are usually referring to “good” or “useful” $q$-analogs, which doesn’t have a widely accepted standard definition, but which I’ll attempt to define here:

More Accurate Definition: An interesting $q$-analog of a statement or expression $P$ is a statement or expression $P_q$ depending on $q$ such that:

1. Setting $q=1$ or taking the limit as $q\to 1$ results in $P$,
2. $P_q$ can be expressed in terms of (possibly infinite) sums or products of rational functions of $q$ over some field,
3. $P_q$ gives us more refined information about something that $P$ describes, and
4. $P_q$ has $P$-like properties.

Because of Property 2, most people would agree that $5^q$ is not an interesting $q$-analog of $5$, because usually we’re looking for polynomial-like things in $q$.

On the other hand, $\frac{q^5-1}{q-1}$, is an excellent $q$-analog of $5$ for a number of reasons. It certainly satisfies Property 2. It can also be easily generalized to give a $q$-analog of any real number: we can define $$(a)_q=\frac{q^a-1}{q-1},$$ a $q$-analog of the number $a$.

In addition, for positive integers $n$, the expression simplifies:
$$(n)_q=\frac{q^n-1}{q-1}=1+q+q^2+\cdots+q^{n-1}.$$
So for instance, $(5)_q=1+q+q^2+q^3+q^4$, which is a natural $q$-analog of the basic fact that $5=1+1+1+1+1$. The powers of $q$ are just distinguishing each of our “counts” as we count to $5$. This polynomial also captures the fact that $5$ is prime, in a $q$-analog-y way: the polynomial $1+q+q^2+q^3+q^4$ cannot be factored into two smaller-degree polynomials with integer coefficients. So the $q$-number $(5)_q$ also satisfies Properties 3 and 4 above: it gives us more refined information about $5$-ness, by keeping track of the way we count to $5$, and behaves like $5$ in the sense that it can’t be factored into smaller $q$-analogs of integers.

But it doesn’t stop there. Properties 3 and 4 can be satisfied in all sorts of ways, and this $q$-number is even more interesting than we might expect. It comes up in finite geometry, analytic number theory, representation theory, and combinatorics. So much awesome mathematics is involved in the study of $q$-analogs that I’ll only cover one aspect of it today: $q$-analogs that appear in geometry over a finite field $\mathbb{F}_q$. Turn to the next page to see them!