Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 17B / c / Solution

The three-term recurrence theorem with the displayed coefficients identifies $P_n$ with the monic orthogonal polynomial $p_n$. Part (b) says its zeros are precisely the eigenvalues of the real symmetric Jacobi matrix $A_n$, and part (a) says those zeros are distinct and all belong to $(-1,1)$.