Chapter 1: Fundamental Theorem of Quantum Calculus

by Wei Xi Fan

Let us take a detour into the world of quantum calculus. <insert weird chromatic 8-bit beeps>

L: It is rather embarrassing that sums are not additive, isn’t it?
R: Hmm…you mean for $a < b < c,$

$\displaystyle \sum_{i=a}^c x_i \neq \sum_{i=a}^b x_i + \sum_{i=b}^c x_i? \qquad \text{(*)}$

L: Yes, precisely. Let’s fix it.
R: OK. How about for $a let’s use

$\displaystyle \sum_{i=a}^b x_i := x_a + x_{a+1} + \ldots + x_{b-1}?$

L: Yes, that works! After this change, we will now have a sum that is additive. To wit, in (*) we now have in the left-hand side terms $x_a + \ldots + x_{c-1},$ and on the right-hand side the last term of the first sum is $x_{b-1},$ while the first term of the second sum is $x_b,$ thus matching up perfectly.
R: What about when $a=b$ or $a
L: Well, to make additivity work, we will have to define

$\displaystyle \sum_{i=a}^a x_i = 0$

when $a=b,$ and

$\displaystyle \sum_{i=a}^b x_i = -\sum_{i=b}^a x_i$

when $a
R: Does additivity still hold with these definitions?
L: Why, yes, this is why we made such definitions in the first place.
R: The fundamental theorem of calculus is nice.
L: Why did you bring that up?
R: Well, it’s nice. It says integration and differentiation are inverse operations.
L: Hmm. Wouldn’t it be nice if there is somehow an inverse operation to summation?
R: Suppose we had a sequence of numbers $x_1, x_2, \ldots$ and we transformed it into a new sequence $y_1, y_2, \ldots,$ where $y_k = \sum_{i=1}^k x_i.$ (Using our new notation, of course.) What can we do to recover the original sequence $x_1, x_2, \ldots?$
L: Well, we can take the successive differences:

$y_2 - y_1, y_3 - y_2, y_4 - y_3, \ldots$

R: Which is?
L: Let’s see…

\begin{aligned} y_2 - y_1 & = x_1 - 0 = x_1; \\ y_3 - y_2 & = (x_1 + x_2) - (x_1) = x_2; \\ y_4 - y_3 & = (x_1 + x_2 + x_3) - (x_1 + x_2) = x_3. \end{aligned}

Oh! This reminds me of the fundamental theorem of calculus. The analogy is that our sequence $x_1, x_2, \ldots$ is like a function $f(x)$ in calculus. When we transform it to a new, accumulative sequence $x_1, x_1 + x_2, x_1 + x_2 + x_3, \ldots,$ this corresponds to us transforming $f(x)$ into another, accumulative function $F(x) = \int_1^x f(t) \; dt.$
R: Exactly. The accumulation occurs discretely for our sequence, while the accumulation is infinitesimal for our function (a.k.a. integration).
L: But what does taking successive difference correspond to?
R: Hint: discrete versus infinitesimal.
L: I understand now. Taking successive differences corresponds to differentiation: given a sequence $(y_i),$ we can make a new sequence $y_2 - y_1, y_3 - y_2, \ldots,$ corresponding to making a new function $g'(x)$ by differentiating some function $g(x)$ at each point.
R: So does summing and then differencing cancel each other out?

$x_1, x_2, \ldots$

Let’s sum it.

$0, x_1, x_1 + x_2, x_1 + x_2 + x_3, \ldots$

Let’s difference it.

$x_1, x_2, \ldots$

We got back our original sequence. This tells us that differencing is the opposite of summation. Wait a minute… that’s what we did a minute ago.
R: If we really waited a minute, this is a tautology. What happens if we difference first, and then sum?
L: Well, let’s start with $x_1, x_2, \ldots$ again and first difference it:

$x_2 - x_1, x_3 - x_2, x_4 - x_3, x_5 - x_4, \ldots$

Let’s sum it. Watch this telescoping:

$0, x_2 - x_1, x_3 - x_1, x_4 - x_1, x_5 - x_1, \ldots$

Hmm…we got our original sequence, except each term had $x_1$ subtracted away from it.
R: What does this remind you of?
L: $\int_a^b f'(t) \; dt = f(b) - f(a).$
R: That’s right: the second fundamental theorem of calculus.
L: Great! We should give the operation of finite difference a symbol so we can write down our fundamental theorems in a succinct form.
R: Given a sequence $x = (x_i)_{i \in \mathbb{Z}},$ let us define a new sequence $\Delta x$ by $(\Delta x)_i = x_{i+1} - x_i.$
L: $\mathbb{Z}?$ Our sequences are double-ended now?
R: Why not? Such sequences are just functions from $\mathbb{Z}$ into $\mathbb{R}$ anyway.
L: Oh. I guess all of the above remains unchanged anyway, so this is fine. Using $i=1$ as a starting point is arbitrary anyway.
R: What are our fundamental theorems?
L:

1. $\displaystyle \left(\Delta \sum_{i=a}^n x_i \right)_j = x_j,$
2. $\displaystyle \sum_{i=a}^b (\Delta x)_i = x_b - x_a,$

corresponding to

1. $\displaystyle \frac{d}{dx} \int_a^x f(t) \; dt = f(x),$
2. $\displaystyle \int_a^b f'(x) \; dx = f(b) - f(a).$

R: Let us call this shadow of the real calculus some kind of umbral calculus.
L: Why not quantum calculus? It is discrete, after all.
R: Why not finite calculus then?