%0 Journal Article %T Chebyshev Polynomials with Applications to Two-Dimensional Operators %A Alfred Wünsche %J Advances in Pure Mathematics %P 990-1033 %@ 2160-0384 %D 2019 %I Scientific Research Publishing %R 10.4236/apm.2019.912050 %X A new application of Chebyshev polynomials of second kind U_n(x) to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows to reduce powers and smooth functions of them to superpositions of the first N-1 powers of the considered operator in N-dimensional case. The method leads in two-dimensional case first to the recurrence relations for Chebyshev polynomials and due to initial conditions to the application of Chebyshev polynomials of second kind U_n(x). Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind U_n(x) comprising the known Generating function as special cases is constructed by means of a derived identity for operator functions f(A) of a general two-dimensional operator A. The basic results are Formulas (9.5) and (9.6) which are then specialized for different examples of functions f(x). The generalization of the theory for three-dimensional operators is started to attack and a partial problem connected with the eigenvalue problem and the Hamilton-Cayley identity is solved in an Appendix. A physical application of Chebyshev polynomials to a problem of relativistic kinematics of a uniformly accelerated system is solved. All operator calculations are made in coordinate-invariant form. %K Hypergeometric Function %K Jacobi Polynomials %K Ultraspherical Polynomials %K Chebyshev Polynomials %K Legendre Polynomials %K Hamilton-Cayley Identity %K Generating Functions %K Fibonacci and Lucas Numbers %K Special Lorentz Transformations %K Coordinate-Invariant Methods %U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=97329