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The spectrum of a new class of Sylvester-Kac matrices

We define a new class of Sylvester-Kac matrices and calculate their spectra
 explicitly. We use the technique of the left eigenvectors to obtain the
 claim. We also provide some right eigenvectors which can be useful in
 applied computations. The main results are rather general and contain many
 known particular characterizations. Matrices belonging to this family
 represent a convenient test matrices for numerical eigenvalue computations
 with known spectrum.