# Homework Help: Chebyshev polynomial - induction problem

1. Dec 13, 2011

### Ryuky

1. The problem statement, all variables and given/known data

Let Tn(x)=cos(narccosx) where x is real and belongs to [-1,1] and n E Z+
Find T1(x).
Show that T2(x)=2x^2 - 1.
Show that Tn+1 (x) + Tn-1 (x) = 2xTn(x)
Hence, prove by induction that Tn(x) is a polynomial of degree n.

3. The attempt at a solution

Since cosθ=x and arccosx=θ

we have: T1(x)=cos(arccosx)=cosθ=x.
T2(x)=cos(2arccosx)=cos2θ=cos^2 θ - sin^2 θ = 2cos^2 - 1 = 2x^2 - 1.

Tn+1 (x) + Tn-1(x) = cos((n+1)arccosx) + cos((n-1)arccosx) = 2cos(narccosx)cos(arccosx)=2Tn(x)*x.

As for the induction we have:

For n=1 T1(x)=x a polynomial of degree 1.
Assuming for n to be true, i.e.Tn(x)=cos(narccosx) is a polynomial of degree n, prove
T(n+1)(x)=cos((n+1)arccosx) is of degree n+1.

So, cos((n+1)arccosx)=cos(nθ + θ)=cosnθ*x - sinnθsinx ... Obviously this is not the way to approach the problem.

Anyway, I would prefer to hear tips from you rather than a complete solution.

2. Dec 13, 2011

### LCKurtz

But you already have Tn+1 (x) + Tn-1 (x) = 2xTn(x) which expresses Tn+1 in terms of lower degree polynomials. Use that.

3. Dec 13, 2011