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?

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.
L: Or we can call them falling factorials. Or falling powers.


About mlbaker

just another guy trying to make the diagrams commute.
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One Response to Chapter 2: Quantum Monomials

  1. Erik Crevier says:

    I practically died laughing @ IGNIS and GRUNT.

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