Polynomials and Sequence Spaces - Wild Linear Algebra - NJ Wildberger

CONTENT SUMMARY: pg 1: @
polynomials and sequence spaces; remark about expressions versus objects @ ;
pg 2: @ 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 @;
pg 4: @ 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 @;
pg 6: @ Forward and backward differences on a sequence; difference below/above convention;
pg 7: @27:21 Forward and backward Differences of lowering powers; calculus reference @;
pg 8: @31:27 Forward and backward Differences of raising powers; operators act like derivative @ ; n equals 0 raising and lowering defined;
pg 9: @ Introduction of some new basis; standard/power basis, lowering power basis, raising power basis; proven to be bases;
pg 10: @ WLA22_pg10_Theorem Newton; proof;
pg 10b: @44:40 Lesson: it helps to start at n=0; example square pyramidal numbers;an important formula @;
pg 11: @50:00 formula of Archimedes; taking forward distances compared to summation @
pg 12: @ a simpler formula; example: sum of cubes;
pg 13: @ exercises 22.1-4;
pg 14: @59:06 exercise 22.5; find the next term; closing remarks @;
Introduction
Some polynomials and associated sequences
Lowering factorial powers
Forward and backward differences
Differences of lowering and raising powers are easy to compute!
Factorial power bases
A theorem of Newton
A formula of Archimedes
A formula for sum of cubes
Exercises 22.1-4;