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I discovered and proved the following theorem back in high school, and have waited patiently to hear something about throughout my college career (which is nearing it's end, hope to have finished my doctorate within the year) to no avail. I am hoping that some of you may recognize the theorem and be able to shed some light on it's history, or even give me it's proper name. Having nothing better to call it I have referred to it as "The fundamental theorem of discrete calculus" due to it's similarity to the fundamental theorem of calculus. Anyway, here is the theorem (presented much more clearly then I did back in high school).

Let $p(x)$ be a polynomial of degree $d$ have the following representation in the binomial basis, $$p(x) = \sum_{i=0}^d a_i \binom{x}{i}$$

Also let $C$ be an arbitrary constant and let$$P(x) = \sum_{i=0}^d a_i \binom{x}{i+1} + C$$

Then $$\sum_{i=a}^b p(i) = P(b+1) - P(a)$$

This is clearly analogous to the fundamental theorem of calculus where $P$ is like a "discrete antiderivative" of $p$.

Also of note is that this trivializes many of the induction proofs that we go through when first learning induction, such as $$\sum_{i=1}^n i = \binom{n+1}{2}$$

So anything you can tell me about the history of the theorem or the subject of "discrete calculus" (which I don't even know if that is the proper name) would be greatly appreciated, thanks in advance!

Elliot
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  • Calculus of finite differences. Also note my question here – abiessu Feb 21 '14 at 23:01
  • The wikipedia article – abiessu Feb 21 '14 at 23:13
  • I seem to recall Melvin Hausner wrote a very readable article on this theorem (a more general version, actually). Try to find it (or ask him, I guess). – msh210 Feb 21 '14 at 23:22
  • Thanks, at least I know it's proper name now and there is plenty of good stuff on the wiki. Clearly I had been using the forward difference throughout my study of this and never even thought of doing it a different way. – Elliot Feb 21 '14 at 23:39
  • So clearly Newton played a major hand in developing the subject, and considering the interpenetration of approximating the continuous derivative it seems likely that he was the first. Can anyone confirm this?

    Also according to wikipedia a major application is in numerical approximation of differential equations. But over the years I have used this (or related ideas) several times while working out problems in real analysis, combinatorics and number theory. So is there a reason this is not taught anywhere but presumably in numerical analysis?

     – Elliot Feb 22 '14 at 02:17

1 Answers1

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Here are some other things to consider.

As a generalization, this identity $$ \sum_{j=m}^{n-k}\binom{n-j}{k}\binom{j}{m}=\binom{n+1}{k+m+1}\tag{1} $$ is proven in this answer using Vandermonde's Identity. If we set $k=0$, we get that $$ \sum_{j=m}^{n}\binom{j}{m}=\binom{n+1}{m+1}\tag{2} $$ from which, your formula is easily verified.

$(2)$ is the inverse of the forward difference of binomial coefficients: $$ \binom{j+1}{m+1}-\binom{j}{m+1}=\binom{j}{m}\tag{3} $$ That is, we can sum $(3)$ in $n$ to get $(2)$ in a telescoping sum. $(3)$ is simply a rearrangement of the identity defining Pascal's Triangle: $$ \binom{j+1}{m+1}=\binom{j}{m}+\binom{j}{m+1}\tag{4} $$

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robjohn
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  • Thanks. I guess I never really thought of it as an identity on binomial coefficients since I was so focused on the polynomials when I first studied this. In fact I originally had everything phrased in the standard basis and had a bunch of ugly coefficients. It took a while for me to recognize them as having the form of the binomial coefficients. – Elliot Feb 23 '14 at 00:26