Toeplitz matrices commuting with tridiagonal matrices

we prove that if R is a nonscalar Toeplitz matrix R_{i, j}=r_?i?j? which commutes with a tridiagonal matrix with simple spectrum, then

$\text{r}{}_{\mathrm{k}}\text{r}{}_1\text{=}\text{u}{}_{\mathrm{k-1}}\text{r}{}_2\text{r}{}_1\text{cos p}\text{u}{}_{\mathrm{k-1}}\text{(cos p)}$

, k=4, 5,…, with U_k the Chebychev polynomial of the second kind, where p is determined from

$\text{cos p=}\text{1}\text{2}\text{r}{}^2{}_1\text{?r}{}_1\text{r}{}_3\text{r}{}^2{}_2\text{?r}{}_1\text{r}{}_3$

.