Polynomials and sequence spaces | Wild Linear Algebra A 22 | NJ Wildberger

Polynomials can be interpreted as functions, and also as sequences. In this lecture we move to considering sequences. Aside from the familiar powers, we introduce also falling and rising powers, using the notation of D. Knuth. These have an intimate connection to forward and backward difference operators. We look at some particular sequences, such as the square pyramidal numbers, from the view of this `difference calculus'.

CONTENT SUMMARY: pg 1: @00:08
polynomials and sequence spaces; remark about expressions versus objects @03:27 ;
pg 2: @04:24 Some polynomials and associated sequences; Ordinary powers; Factorial powers (D. Knuth);
pg 3: @10:34 Lowering (factorial) power; Raising (factorial) power; connection between raising and lowering; all polynomials @13:28;
pg 4: @13:52 Why we want these raising and lowering factorial powers; general sequences; On-line encyclopedia of integer sequences (N.Sloane); 'square pyramidal numbers'; Table of forward differences;
pg 5: @19:23 Forward and backward differences; forward/backward difference operators on polynomials; examples: operator on 1 @23:07;
pg 6: @23:38 Forward and backward differences on a sequence; difference below/above convention;
pg 7: @27:21 Forward and backward Differences of lowering powers; calculus reference @29:37;
pg 8: @31:27 Forward and backward Differences of raising powers; operators act like derivative @34:45 ; n equals 0 raising and lowering defined;
pg 9: @36:17 Introduction of some new basis; standard/power basis, lowering power basis, raising power basis; proven to be bases;
pg 10: @39:23 WLA22_pg10_Theorem (Newton); proof;
pg 10b: @44:40 Lesson: it helps to start at n=0; example (square pyramidal numbers);an important formula @47:47;
pg 11: @50:00 formula of Archimedes; taking forward distances compared to summation @52:46
pg 12: @53:20 a simpler formula; example: sum of cubes;
pg 13: @57:38 exercises 22.1-4;
pg 14: @59:06 exercise 22.5; find the next term; closing remarks @59:50;

Video Chapters:
00:00 Introduction
4:23 Some polynomials and associated sequences
10:32 Lowering (factorial) powers
19:22 Forward and backward differences
27:20 Differences of lowering and raising powers are easy to compute!
36:16 Factorial power bases
39:23 A theorem of Newton
49:58 A formula of Archimedes
53:20 A formula for sum of cubes
57:38 Exercises 22.1-4;
************************

Here are the Insights into Mathematics Playlists:

[gucker]

Thank you very much! I am overwhelmed by the fact that it is possible (and so easy!) to restore a polynomial from a (polynomially generated) sequence. This material is so beautiful! Such a pity that it is not taught in any standard mathematical courses. Thank you, professor Wildberger!!

[onepunchtaco]

This is beautiful Prof. you are awesome

[njwildberger]

Hi comecra85, I will be talking about bilinear and quadratic forms in a few more lectures, but tensors are a bit too advanced for this course.

[njwildberger]

You are right, one of the Bernoulli's found first the formula for sums of powers.

[Ghostly]

if you wrote a book, i would be the first in line to buy it! So much better than my professor and my stiff textbooks!!!

[rummanmuhammad]

Hello doc .. how r u ? very nice presentation

[aynurelbistan]

Hello, it was really helpful to me.I have some question about that.How can I contact with you?