For any positive integer n, use de Moivre's formula and the binomial theorem to show that the nth Chebyshev polynomial i

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For Any Positive Integer N Use De Moivre S Formula And The Binomial Theorem To Show That The Nth Chebyshev Polynomial I 1 (64.14 KiB) Viewed 63 times

For any positive integer n, use de Moivre's formula and the binomial theorem to show that the nth Chebyshev polynomial is: [n/2] n Σ (2) 2-¹ (2-1). 1=0 Pn(z) = [ where [n/2] denotes the largest integer less than or equal to n/2. Continue "simplifying" to obtain the following alternative, more explicit formula for pn(z): [n/2] [n/2] zn-2k Pn(2) = [(-1)² Σ ( 2 ) ( ) ] ²-². k=0 1=k