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To explain the similarity between the Leibniz formula and the binomial theorem, one answer by Tad uses paths to justify the similarity in the coefficients of the two formulae.
I understand the answer completely, but I want to know whether anyone knows an academic source for the method used in this answer. Or is this answer original?
Thanks!
P.S. I wanted to comment on the answer itself, but I'm new to StackExchange and do not have enough reputation for commenting on the answer.

Edit: I'm not looking for any elaboration of the relationship between the two formulae; I'm just looking for whether such a "path-based" approach has been used earlier in existing literature i.e. for citation purposes in a research paper

2 Answers2

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You can check out the wikipedia page: https://en.wikipedia.org/wiki/General_Leibniz_rule.

You will find that the proof by induction of Leibniz formula is almost exactly the same as the binomial formula, and it is a purely algebraic proof in the sense that it is just about manipulating sums and products.

But just like you don't remember and understand the binomial formula by remembering the proof by induction, I believe the best way to think about Leibniz formula is to understand that it is merely a sum over all the possible paths obtained by choosing which one of $f$ or $g$ you take the derivative in a series of $n$ possibilities. The post you refer to gives an excellent explanation of this. Hope this helps!

DodoDuQuercy
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  • Hi! Thanks for the information, but I'm looking for whether there is an academic reference for such a "path-based" approach in existing literature. i.e. which I can use to cite in a research paper. –  Jun 21 '20 at 12:09
  • Oh okay! Sorry I don't know of any academic reference to this, hope you manage to find one! – DodoDuQuercy Jun 21 '20 at 12:18
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  • This path based approach is often used in combinatorics. You might want to skim through Analytic Combinatorics by R. Sedgewick and P. Flajolet to find many examples.

  • The calculus of finite differences plays a key role here and you might have a look at the umbral calculus which provides a glueing theory between the binomial theorem and relationships following this binomial type. A classic is The Umbral Calculus by S. Roman.

Markus Scheuer
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  • Thanks! These are great resources, but none of them contain the specific proof in the answer I was referring to. It would be great if you know whether that specific proof is present in existing literature... But thanks! –  Jun 21 '20 at 16:16