Question 24
For $n \ge 1$, let $P_n(x)$ be the polynomial defined by $P_1(x) = 1$, $P_2(x) = 2x$, and
$$
P_n(x) = 2x P_{n-1}(x) - (x^2 + 1) P_{n-2}(x)
$$
for $n \ge 3$. Prove that we have $|P_n(x)| \le (1 + x^2)^{n/2}$ for any positive integer $n$ and real number $x$, and determine the conditions under which equality holds.
$$
P_n(x) = 2x P_{n-1}(x) - (x^2 + 1) P_{n-2}(x)
$$
for $n \ge 3$. Prove that we have $|P_n(x)| \le (1 + x^2)^{n/2}$ for any positive integer $n$ and real number $x$, and determine the conditions under which equality holds.