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 + q² + q³ + … + qn-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 – qa) / (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
limq → 1 [a]q = a.
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! = q q 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.)
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.
2 thoughts on “Mathematical plot twist: q-binomial coefficients”
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