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
L: Yes, precisely. Let’s fix it.
R: OK. How about for let’s use
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 and on the right-hand side the last term of the first sum is while the first term of the second sum is thus matching up perfectly.
R: What about when or
L: Well, to make additivity work, we will have to define
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 and we transformed it into a new sequence where (Using our new notation, of course.) What can we do to recover the original sequence
L: Well, we can take the successive differences:
R: Which is?
L: Let’s see…
Oh! This reminds me of the fundamental theorem of calculus. The analogy is that our sequence is like a function in calculus. When we transform it to a new, accumulative sequence this corresponds to us transforming into another, accumulative function
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 we can make a new sequence corresponding to making a new function by differentiating some function at each point.
R: So does summing and then differencing cancel each other out?
L: Well let’s see. Let’s start with
Let’s sum it.
Let’s difference it.
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 again and first difference it:
Let’s sum it. Watch this telescoping:
Hmm…we got our original sequence, except each term had subtracted away from it.
R: What does this remind you of?
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 let us define a new sequence by
L: Our sequences are double-ended now?
R: Why not? Such sequences are just functions from into anyway.
L: Oh. I guess all of the above remains unchanged anyway, so this is fine. Using as a starting point is arbitrary anyway.
R: What are our fundamental theorems?
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?