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Substitute $D_N={\beta }^N$ into the recurrence. Since $k>0$, its roots are nonzero and satisfyFor distinct roots, the general recurrence solution is $D_N=c_1{\beta }_1^N+c_2{\beta }_2^N$. The initial values require $c_1+c_2=1$ and $c_1{\beta }_1+c_2{\beta }_2=d={\beta }_1+{\beta }_2$, givingAt a repeated root, $d=\pm 2k$ and ${\beta }_1={\beta }_2=\beta =\pm k$, this quotient must be replaced by its limit:It also follows by solving the repeated-root recurrence as $(c_1+c_{2}N){\beta }^N$. This exceptional case is necessary to avoid introducing a false root in the next part; it is included in the fixed-end spring-chain determinant and its repeated-root limit.

For distinct nonzero recurrence roots, the determinant formula givesThe ratio one is excluded: at that repeated root, part (iii) gives $(N+1)(\pm k{)}^N\ne 0$. Thus the naive extra root corresponding to equal recurrence roots is spurious.

To count the physical solutions without ambiguity from exchanging ${\beta }_1$ and ${\beta }_2$, write ${\beta }_{1,2}=k{e}^{\pm iq}$ and $d=2k\cos q$ with $0<q<\pi$. ThenIts $N$ distinct zeros occur at $q=j\pi /(N+1)$, $j=1,\dots ,N$. They give exactly $N$ distinct positive normal frequencies, or equivalently $N$ distinct roots in ${\omega }^2$:If signed $\omega$ is counted, the characteristic equation has $2N$ roots $\pm {\omega }_j$. The sine quotient is the Chebyshev polynomial of the second kind representation of the fixed-end spring-chain determinant and its repeated-root limit.

For $N$ identical masses between fixed walls with equal springs, the dimensionless stiffness matrix has diagonal entries $2$ and adjacent entries $-1$. Its normalized eigenvectors and eigenvalues areExtend each vector by $a_0=a_{N+1}=0$. The sine addition identity gives $2a_i-a_{i-1}-a_{i+1}={\lambda }_j{a}_i$, including the fixed-end rows. The finite sine orthogonality identity supplies the normalization and makes distinct vectors orthogonal. The normal mode angular frequencies are ${\omega }_j=\sqrt{k{\lambda }_j/m}$ for mass $m$ and spring constant $k$. This complements the fixed-end spring-chain determinant and its repeated-root limit with the spatial mode shapes.