Polynomials and sequence spaces | Geometric Linear Algebra 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; ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.