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Define a sequence of polynomials in \(\mathbb{R}[X]\) by $$ P_0 = 0, \qquad P_1 = 1, \qquad P_{n+1} = X P_n + P_{n-1} \quad (n \geq 1). $$
  1. Compute \(P_2\), \(P_3\), \(P_4\), \(P_5\).
  2. Show that \(P_n \wedge P_{n+1} = 1\) for every \(n \in \mathbb{N}\).
  3. Deduce that there exist \(U_n, V_n \in \mathbb{R}[X]\) such that \(P_n U_n + P_{n+1} V_n = 1\).

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