1.7.2 Example B 49 Prove the following (Newton's theorem): If fk is a polynomial of the nth degree, then it can be written in the form fk = fo + Afo 1(1) + A-Jo k(2) + .. A" fo 1(1). (1.186) 1! 2! n!...

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Extracted text: 1.7.2 Example B 49 Prove the following (Newton's theorem): If fk is a polynomial of the nth degree, then it can be written in the form fk = fo + Afo 1(1) + A-Jo k(2) + .. A" fo 1(1). (1.186) 1! 2! n! Assume that fk has the representation fk = ao + a1k(1) + a2k(2) +… .. + ank(n), (1.187) where ao, a1,..., an are constants. Differencing fk n times gives = a1 + 2a2k(1) + 3azk(2) + ·.. + nank(n-1). A² fk = 2·1. a2 + 3· 2. azk) +.. + n(n – 1)a,k(n–2), (1.188) A" fk = ann(n – 1)· . · (1). THE DIFFERENCE CALCULUS 25 Setting k = 0 in the original function and its differences allows us to conclude that A™ fo Am = m = 0,1,...., n. (1.189) т! To illustrate the use of this theorem, consider the function fk = k4. (1.190) Now Afk = 4k3 + 6k² + 4k + 1, A² fk = 12k? + 24k + 14, (1.191) A³ fk = 24k + 36, Aª fk = 24, and