## Chapter 2: Quantum Monomials

by Wei Xi Fan

R: What is $\frac{d}{dx} x^n?$
L: $nx^{n-1}.$
R: What is the quantum analogue of $x^n?$
L: I have no clue.
R: Let’s try $i^n.$ This is the sequence $1^n, 2^n, 3^n, \ldots$
L: OK. $(\Delta i^n)_j = (j+1)^n - j^n$. I can’t get anywhere from here that would make it look remotely close to $nj^{n-1}$.
R: What do we do?
L: We call upon the fire of inspiration itself, Ignis.
IGNIS: The truth ye seek is but a cascading staircase, but one that ends.
L: What do you think that means?
R: Instead of trying $i^n = i* i * i * \ldots * i$, let’s try

$i^{[n]} := i * (i-1) * \ldots * (i-n+1)$

there are $n$ terms in total just like $i^n$, but here, each term is one less than the previous.
L: Like a factorial that suddenly ends. All right, so

\begin{aligned} (\Delta i^{[n]})_j &= (i+1)^{[n]} - i^{[n]} \\ &= (i+1) * i * (i-1) * \ldots * (i-n+2) - i * (i-1) * \ldots * (i-n+2) * (i-n+1) \end{aligned}

Oh! The factors line up. The subtraction yields

$i * (i-1) * (i-n+2) * [i+1-(i-n+1)] = n * i^{[n-1]}.$

This is exactly what we are looking for.
R: Great! So we have the identity

$(\Delta i^{[n]})_j = n * j^{[n-1]} \qquad (*)$

corresponding to

$\displaystyle \frac{d}{dt} x^n = nx^{n-1}$

in the real calculus.
L: OK. What about when $n=0$ or when $n < 0$?
R: Well… what do we do to get from $x^{[n+1]}$ to $x^{[n]}?$
L: Well, we divide $x^{[n+1]}$ by its last term, which is $(x-n)$, to get to $x^{[n]}$, because the only difference between them is that $x^{[n]}$ is missing the $(x-n)$ term.
R: What do we get when we go from $x^{[1]}$ to $x^{[0]}?$
L: Well, $x^{[1]} = x$, and so we divide this by $(x-0)$ to get to $1$, so I guess we should define $x^{[0]} := 1.$
R: What about $x^{[-1]}?$
L: Well, we would divide this by $(x+1),$ so I guess it is $1/(x+1).$
R: So what’s the general formula?
L:

$x^{[n]} = \begin{cases} x(x-1)(x-2) \ldots (x-n+1) & \text{if } n > 0, \\ 1 & \text{if } n = 0, \\ 1/(x+1)(x+2)\ldots(x+(-n)) & \text{if } n < 0. \end{cases}$

Are you sure the identity (*) still holds for these new definitions?
R: For these situations, we should call upon the minion of non-illuminating bashing, Grunt.
GRUNT: ‘Tis done.
R: There, this fact has been verified.
L: I shall call these Pochhammer symbols.
R: WTF?
L: Or we can call them falling factorials. Or falling powers.