The simplest instance of a *q*-analog in math is to replace a positive integer *n* by a polynomial in *q* that converges to *n* as *q* → 1. Specifically, we associate with *n* the polynomial

[*n*]_{q} = 1 + *q* + *q*² + *q*³ + … + *q*^{n-1}.

which obviously converges to *n* when *q* → 1.

More generally we can associate with a real number *a* the rational polynomial

[*a*]_{q} = (1 – *q*^{a}) / (1 – *q*).

When *a* is a positive integer, the two definitions are the same. When *a* is not an integer, you can see by L’Hôpital’s rule that

lim_{q → 1} [*a*]_{q} = *a*.

## Motivation

Why would you do this? I believe the motivation was to be able to define a limit of a series where the most direct approach fails or is difficult to work with. Move over to the land of *q*-analogs, do all your manipulations there where requiring |*q*| < 1 avoids problems with convergence, then take the limit as *q* goes to 1 when you’re done.

## q-factorials and q-binomial coefficients

The factorial of a positive integer is

*n*! = 1 × 2 × 3 × … × *n*

The ** q-factorial** of

*n*replaces each term in the definition of factorial with its

*q*-analog:

[*n*]_{q}! = [1]_{q}! [2]_{q}! [3]_{q}! … [*n*]_{q}!

(It helps to think of []_{q}! as a single operator defining the *q*-factorial as above. This is not the factorial of [*n*]_{q}.

Similarly, you can start with the definition of a binomial coefficient

*n*! / (*k*! (*n*–*k*)!)

and define the ** q-binomial** coefficient by replacing factorials with

*q*-factorials.

[*n*]_{q}! / ([*k*]_{q}! [*n*–*k*]_{q}!)

You can apply the same process to rising and falling powers, and then replace the rising powers in the definition of a hypergeometric function to define *q*-analogs of these functions.

(The *q*-analogs of hypergeometric functions are confusingly called “basic hypergeometric functions.” They are not basic in the sense of simple examples; they’re not examples at all but new functions. And they’re not basic in the sense of a basis in linear algebra. Instead, the name comes from *q* being the base of an exponent.)

## Plot twist

And now for the plot twist promised in the title.

I said above that the *q*-analogs were motivated by subtle problems with convergence. There’s the idea in the background that eventually someone is going to let *q* approach 1 in a limit. But what if they don’t?

The theorems proved about *q*-analogs were first developed with the implicit assumption that |*q*| < 1, but at some point someone realized that the theorems don’t actually require this, and are generally true for any *q*. (Maybe you have to avoid *q* = 1 to not divide by 0 or something like that.)

Someone observed that *q*-analog theorems are useful when *q* is a prime power. I wonder who first thought of this and why. It could have been inspired by the notation, since *q* often denotes a prime power in the context of number theory. It could have been as simple as a sort of mathematical pun. What if I take the symbol *q* in this context and interpret it the way *q* is interpreted in another context?

Now suppose we have a finite field *F* with *q* elements; a theorem says *q* has to be a prime power. Form an *n*-dimensional vector space over *F*. How many subspaces of dimension *k* are there in this vector space? The answer is the *q*-binomial coefficient of *n* and *k*. I explore this further in the next post.

Weirdly, I believe Gauss introduced the variable q for basic hypergeometric functions before the connection to prime powers was noticed – or, of course, the connection to quantum groups, where q means “quantum”.

Never heard of q-analogs before. Had to consult Wikipedia and ended up with the field with one element https://en.wikipedia.org/wiki/Field_with_one_element

Nice.