coefficient by $x^{n-1}$ in Chebyshew polynominals

Question

Calculate coefficient by $x^{n-1}$ in Chebyshew polynominal of the first kind $T_n$, defined as: $$ T_0(x)=1\\ T_1(x)=x\\ T_n(x)=2x\cdot T_{n-1}(x)-T_{n-2}(x) $$

Answer 1

The coefficient by $x^{n-1}$ of $T_n(x)$ is $0$. One can use induction:

• $n=0$ and $n=1$ are clear.
• Assume it is true for $0\leq m <n$. Let $a$ be the coefficient by $x^{n-2}$ of $T_{n-1}(x)$. By the induction hypothesis, $a=0$ From the expression, $T_n(x)=2xT_{n-1}-T_{n-2}(x)$, we know that the coefficient by $x^{n-1}$ of $T_n(x)$ is $2a$, which is $0$, as wanted.

Note: For this to work you should also prove that $T_n(x)$ has degree $n$, which may also be done by induction.

Answer 2

Hint: Prove by induction that the degree of $T_n$ is $n$ and that the coefficient of $x^{n-1}$ in $T_n$ is zero.

Actually, $T_n$ has only even powers when $n$ is even and only odd powers when $n$ is odd.