Does this formula holds true for $|x|>1$?

Question

When the first two Chebyshev polynomials $T₀(x)$ and $T₁(x)$ are known, all other polynomials $T_{n}(x),n≥2$ can be obtained by means of the recurrence formula $$T_{n+1}(x)=2xT_{n}(x)-T_{n-1}(x)$$

My question is:

Does this formula holds true for $|x|>1$?

Answer 1

The recurrence generates new polynomials, and polynomials are always defined for all $x \in \Bbb R$.

So, yes, it also holds for $|x| > 1$.