Question

TN(-x)=(-1)^NTN(x).

(a) True

(b) False

This question was addressed to me in class test.

My question is from Chebyshev Filters in section Digital Filters Design of Digital Signal Processing

Answer 1

Right option is (a) True

Best explanation: We know that a chebyshev polynomial of degree N is defined as

TN(x) = cos(Ncos^-1x), |x|≤1

cosh(Ncosh^-1x), |x|>1

=> TN(-x)= cos(Ncos^-1(-x))=cos(N(π-cos^-1x))=cos(Nπ-Ncos^-1x)=(-1)^N cos(Ncos^-1x)=(-1)^NTN(x)

Thus we get, TN(-x)=(-1)^NTN(x).