What is Differenciation and How Does It Verify Finite Polynomials?

[Eynbanoiqvs]

The above is not a spelling mistake as I am referring to differenciation rather than differentiation. As to the best of my knowledge, no one else has used the same term nor developed a similar method; and so I claim it as my own till challenged.
Using differenciation, one can verify the expression of any finite polynomial.

I put a lot of work in trying to write up an explanation for my method...so it's best seen in the word document attached. But I still don't think it's perfect.
Please post your views and understanding of this. I would like any feedback.

 

[Hurkyl]

Using differenciation, one can verify the expression of any finite polynomial.

I can't figure out what this sentence is supposed to mean.
I haven't tried downloading your zip file. I'm guessing you're reproducing the theory of difference equations, or possibly have rediscovered some form of Newton series.

 

[Eynbanoiqvs]

Yes; it is the Newton series...but in a primitive form. Thanks for telling me. I didn't know how to search for it or identify it.

That sentence means that you can use the Newton series method to check the formula for any function; assuming it is a finite polynomial and has rational inputs and outputs.
For example, the sum of natural numbers. If one didn't know any theory behind the derivation; this method could yield n^2 /2 +n/2 by calculation with minimum logic involved.
So simple that a computer could derive the formula.

Thanks again for identifying it! I was hoping I was the first...but I guess Newton bet me! :)

 

[Hurkyl]

I've looked at the article, I'll say it's much better written than I expected from a *.doc file posted on the internet!

There's no better way to understand (and to eventually further) a subject than to derive it for yourself, so hopefully you'll continue your study / research, and a lead on existing knowledge will surely help. Sometimes just having good notation makes all the difference!

If nothing else, I think differences are fun -- and the fact of analogies with differentials is interesting -- although I've only spent a little bit of time with them. And they certainly can be very useful in discrete math.

 

[Eynbanoiqvs]

Thanks! I actually used the latest Word 2007 to design the document, and then converted it into the old format for uploading.