Cyclical matrices and the Chebyshev polynomials

A sequence of polynomials generated by a cyclical Jacobi matrix is treated as a function of an integer argument, which is the order of the polynomials. Formulas for the sum and difference of the functions and their arguments that generalize similar formulas for trigonometrical functions are constructed. An expression for such sequences of polynomials with the use of Chebyshev polynomials of the second kind is obtained. A divisibility formula for Chebyshev polynomials of the second kind is obtained. A solution of the inverse problem for Chebyshev polynomials, i.e. a description of the corresponding functions of integer arguments by using their properties is presented.