The eigenvalues of r-periodic tridiagonal matrices by factorization of some recursive sequences

We introduce r-periodic tridiagonal matrices for given integer r≥2. In which the entries on the principle diagonal can be any r-periodic sequence. If the entries on the principle diagonal are equal then the calculation of the eigenvalues of corresponding tridiagonal matrices is relatively easy, but when the entries are not equal, the calculation becomes much more difficult. So, some explicit formulas could be given only for the eigenvalues of certain types of the 2-periodic tridiagonal matrices so far. We give a new algorithm to find the eigenvalues of certain r-periodic tridiagonal matrices and give some results by implementing it in a symbolic programming language. Our algorithm also finds the zeros of some families of polynomials with integer coefficients. The degree of these polynomials can be chosen very high. Also, we give some new properties of the generalized continuant and then we solve an open problem by determining a complex factorization of certainr-periodic sequences. Finally, we generalize existing results by giving the explicit formulas for the eigenvalues of some r-periodic tridiagonal matrices for r=2,3 and 4.