TITLE:
Chebyshev Polynomials with Applications to Two-Dimensional Operators
AUTHORS:
Alfred Wünsche
KEYWORDS:
Hypergeometric Function, Jacobi Polynomials, Ultraspherical Polynomials, Chebyshev Polynomials, Legendre Polynomials, Hamilton-Cayley Identity, Generating Functions, Fibonacci and Lucas Numbers, Special Lorentz Transformations, Coordinate-Invariant Methods
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.12,
December
24,
2019
ABSTRACT: A new application of Chebyshev polynomials of second kind Un(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 Un(x). Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind Un(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.