Question

What is the formula for chebyshev polynomial TN(x) in recursive form?

(a) 2TN-1(x) – TN-2(x)

(b) 2TN-1(x) + TN-2(x)

(c) 2xTN-1(x) + TN-2(x)

(d) 2xTN-1(x) – TN-2(x)

This question was addressed to me in final exam.

This intriguing question originated from Chebyshev Filters topic in portion Digital Filters Design of Digital Signal Processing

Answer 1

Right option is (d) 2xTN-1(x) – TN-2(x)

To explain: We know that a chebyshev polynomial of degree N is defined as

TN(x) = cos(Ncos^-1x), |x|≤1

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

From the above formula, it is possible to generate chebyshev polynomial using the following recursive formula

TN(x)= 2xTN-1(x)-TN-2(x), N ≥ 2.