The Binomial Chu Vandermonde Identity: a new unification? | Algebraic Calculus Two | Wild Egg Maths

We suggest a novel unification of the Binomial and Chu Vandermonde identities, leading to an unusual introduction of the exponential polyseries, along with Newton's reciprocal polyseries.

The main idea is to introduce a generalization of Knuth's rising and falling powers notation, which we call ladder powers, and to extend our inductive proof of the Chu Vandermonde relation to this more general situation. A major consequence of this analysis is that the exponential polyseries, which of course plays a hugely central role in Calculus, is now introduced in the context of a variant of Newton's generalized binomial theorem, (or Newton's polyseries theorem).

We will see that this point of view provides a solid new foundation for a rethinking of several fundamental formulas in elementary Calculus relating to power series.

NOTE: On the last slide 10, there is an incorrect (r+s)! in the denominator instead of n!.

Video Contents (thanks to Lucas Lofaro)

0:00 Intro
0:13 Overview
0:28 Review of Related Identities
4:09 Ladder Powers
6:29 Binomial Chu-Vandermonde Identity
8:57 Binomial C-V Identity for Degree 2
11:36 Binomial C-V Identity for Degree 3
14:47 Connecting Newton Polyseries Theorem with t = -1
18:19 Exponential Polyseries and t = 0
23:36 Exponential Polyseries Identity
26:22 Newton Reciprocal Polyseries and t = 1
29:32 Introducing Reciprocal Polyseries

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[accountname]

It's so interesting that the notation for binomial coefficients obfuscates this relationship! I wonder what other notation has stopped underlying patterns from being spotted. I've always been extremely fascinated by the power of generating functions, they seem to "know" everything... Fantastic and insightful as usual Norm!

[EdEmJuPe]

Whoo hoo! Didn't have to wait 6 months for the new video :)

[steffspinner]

Usuing SCALA, there is an other Library for electronic Hardwar development, which is called "CHISEL", from University Berkeley, enabling to develop new hardware in a more efficient way as today usage of HHDL and Verilog. In combination with the flexible and efficient Type Sytem of SCALA it schould be possible do develop new harware including new processor structures as well as developping processors, that are based on the logic of Boole instead of Shannon or Boolean Algebra, but based of the Principles of polynomial calculations using finite fields.

[steffspinner]

My idea, combining Chisel and Rings in SCALA including some more stuff yet to be developped, this could be very interesting and powerfull!!!

[joseagapito]

Hi Norman, when you introduce (1-\alpha>^-r using raising powers, I think there is a typo when changing r+1 to -r+1 (you wrote up -r-1) and so on.

[sacul]

Video Contents

0:00 Intro
0:13 Overview
0:28 Review of Related Identities
4:09 Ladder Powers
6:29 Binomial Chu-Vandermonde Identity
8:57 Binomial C-V Identity for Degree 2
11:36 Binomial C-V Identity for Degree 3
14:47 Connecting Newton Polyseries Theorem with t = -1
18:19 Exponential Polyseries and t = 0
23:36 Exponential Polyseries Identity
26:22 Newton Reciprocal Polyseries and t = 1
29:32 Introducing Reciprocal Polyseries