= Fixed-end spring-chain determinant and its repeated-root limit
{title2=$D_N=k^NU_N((2k-m\omega^2)/(2k))$}
For $N$ equal masses joined to fixed walls by equal springs, the stiffness matrix has diagonal $2k$ and adjacent entries $-k$. Its characteristic <determinant> obeys $D_N=dD_{N-1}-k^2D_{N-2}$, $d=2k-m\omega^2$, $D_0=1$, $D_1=d$. If $\beta_1+\beta_2=d$ and $\beta_1\beta_2=k^2$, then $D_N=(\beta_1^{N+1}-\beta_2^{N+1})/(\beta_1-\beta_2)$ for distinct roots. At a repeated root $\beta=\pm k$, the correct limit is $(N+1)\beta^N$, which is nonzero. Thus the apparent ratio solution $\beta_1/\beta_2=1$ must be excluded. Writing $d=2k\cos q$ gives $D_N=k^N\sin[(N+1)q]/\sin q$, the <Chebyshev polynomial of the second kind>. There are exactly $N$ distinct positive normal frequencies $2\sqrt{k/m}\sin[j\pi/(2(N+1))]$, $1\leq j\leq N$.
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