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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 $(-1,1)$.

Let $L$ be the symmetric zero-diagonal Jacobi matrix with $L_{j,j+1}=L_{j+1,j}=a_j$, and let the skew-symmetric matrix $B$ haveThen $\dot{L}=[B,L]$ is equivalent to the Finite Volterra lattice. The corresponding similarity flow makes the eigenvalues of $L$, the coefficients of its characteristic polynomial, and its trace invariants first integrals.