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<b, 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<b?
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<b.
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?
L: Well let’s see. Let’s start with

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?

  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?


About mlbaker

just another guy trying to make the diagrams commute.
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One Response to Chapter 1: Fundamental Theorem of Quantum Calculus

  1. This was done in different notation here. and here

    Each of them use function notation instead of sequence notation but it doesn’t make a difference since a sequence can just be viewed as a function on \mathbb{Z}

    Can’t find a constant nomenclature for it though,

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