Recurrence relations for the even and odd characteristic polynomials of a symmetric Toeplitz matrix

In a recent paper (Computation of the smallest even and odd eigenvalues of a symmetric positive-definite Toeplitz matrix, SIAM J. Matrix Anal. Appl. 25 (2004) 949-963) Melman proved a recurrence relation of the even and odd characteristic polynomials of a real symmetric Toeplitz matrix T on which a symmetry exploiting method for computing the smallest eigenvalue of T can be based. In this note, we present a proof of the recurrence relation which is less technical and more transparent.