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.
References
[1]
Szegö, G. (1959) Orthogonal Polynomials. 2nd Edition, American Mathematical Society, New York.
[2]
Erdélyi, A. (1953) Higher Transcendental Functions, Volume II. McGraw-Hill, New York.
[3]
Abramowitz, M. and Stegun, I. (1965) Handbook of Mathematical Functions. Dover Publication, New York.
[4]
Nikiforov, A.F. and Uvarov, V.B. (1974) Osnovy teorii spetsialnykh funkzij (Fundaments of the Theory of Special Functions). Nauka, Moskva.
[5]
Koornwinder, T.H., Wong, R., Koekoek, R. and Swarttouw, R.F. (2019) Orthogonal Polynomials. In: NIST Handbook, Chapter 18, NIST, Gaithersburg, MD, 435-484.
[6]
Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark C.W. (2010) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge.
[7]
Gradshteyn, I.S. and Ryzhik, I.M. (1963) Tables of Series, Products and Integrals. 4th Edition, Nauka, Moscow (Translation into Other Languages and Later Editions Exist).
[8]
Rivlin, T. (1990) Chebyshev Polynomials, From Approximation Theory to Algebra and Number Theory. 2nd Edition, John Wiley and Sons, New York.
[9]
Erdélyi, A. (1953) Higher Transcendental Functions, Volume I. McGraw-Hill, New York.
[10]
Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus. Academic Press, New York.
[11]
Wünsche, A. (2015) Operator Methods and Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials. Applied Mathematics, 7, 213-261.
https://doi.org/10.4236/apm.2017.72012
[12]
Wünsche, A. (2015) Generating Functions for Products of Laguerre 2D and Hermite 2D Polynomials. Applied Mathematics, 6, 2142-2168.
https://doi.org/10.4236/am.2015.612188
[13]
Conway, J.H. and Guy, R.K. (1996) The Book of Numbers. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-4072-3
[14]
Whittaker, E.T. and Watson, G.N. (1927) A Course of Modern Analysis. 4th Edition, Clarendon Press, Cambridge.
[15]
Akhieser, N.I. (1963) Theory of Approximation. MacMillan Company, New York.
Aleksandrov, A.D., Kolmogorov, A.N. and Lavrent’ev, M.A. (1999) Mathematics Its Content, Methods and Meaning. Dover Publications, Mineola, NY.
[18]
Rivlin, T. (1981) An Introduction to the Approximation of Functions. Dover Publication, New York.
[19]
Gantmakher, F.R. (1954) Teoriya Matrits. Gostekhisdat, Moskva. [In German: Gantmacher, F.R. (1958) Matrizenrechnung, Dtsch. Verl. d. Wissenschaften, Berlin. In English: Gantmacher, F.R. (1960) The Theory of Matrices. Chelsea Publishing, New York.]
[20]
Mirski, L. (1955) An Introduction to Linear Algebra. Clarendon Press, Oxford.
[21]
Fyodorov, F.I. (1958) Optika anisotropnykh sred (Optics of Anisotropic Media), (Optic of Ansiotropic Media). Isdatel’stvo AN BSSR, Minsk.
[22]
Fock, V.A. (1966) The Theory of Space, Time and Gravitation. 2nd Edition, Pergamon Press, Oxford [Russian Original: Fizmatgiz, Moskva (1961).]
