An explicit formula for singular values of the Sylvester–Kac matrix

The object of our interest is a certain tridiagonal matrix that appears in a variety of problems in statistical mechanics and quantum physics, such as the Brownian motion, random walk on a hypercube, the Ehrenfest urn model, and the Stark effect of the hydrogen atom. The spectral decomposition of this matrix has been studied by a number of authors, among others Sylvester, Cayley, Mazza, Muir, Schrödinger, and Kac. In particular, explicit expressions are known for the eigenvalues and the eigenvectors of the matrix. So the question arises: Does there exist an explicit formula for the singular values? In this paper we find an explicit formula for a subset of the singular values when the order of the matrix is odd. In the process we utilize the method of generating functions, and derive a second-order differential equation. The polynomial solutions of this differential equation provide the elements of the singular vectors.