On the zeros of certain linear combinations of Chebyshev polynomials

We consider a system of masses and springs, such that all springs have equal stiffness K, except those springs that possibly connect the boundary masses to earth, and such that all masses except the first and the last are equal to M. The eigenfrequencies of this four parameter system can be written in terms of the zeros of certain linear combinations of Chebyshev polynomials. The physical behaviour of the system suggests sufficient conditions for the parameters such that these zeros are all real. It will be shown by Sturmʹs theorem that under these conditions the eigenvalues of the matrix of the system are real and ⩾0, which implies the conjecture about the zeros. When some additional conditions for the parameters are satisfied, we even can conclude that all the zeros are located in the interval (−1, 1).