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: @00:08polynomials 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;
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