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Divided Differences Calculus in Matrix Representation

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A systematic description of actions of the divided differences operators on power and exponential functions is given. The results of actions of these operators on entire functions are presented by the matrices whose elements are functions of coefficients of a characteristic (pivot) polynomial. Effective algorithms of calculation of the matrices are constructed using the properties of the companion matrix of the pivoting polynomial. Degeneration of the roots of the pivot polynomial reduces the n-order divided differences operator to \(n-1\) order operator of differentiation. The exponential type invariant functions with respect to higher order derivatives are constructed.

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  1. 1.

    Dikoussar, N.D.: Method of basic elements. Math. Models Comput. Simulations 3(4), 492–507 (2011)

  2. 2.

    Carl, De Boor: Divided Differenes. Surv. Approx. Theory 1, 46–69 (2005)

  3. 3.

    Datttoli, G., Licciardi, S., Sabia, E.: Genralized trigonometric functions and matrix parameterization. Int. J. Appl. Comput. Math. 3(Suppl 1), 115–128 (2017)

  4. 4.

    Yamaleev, R.: Complex algebras on N-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics. Adv. Appl. Cliff. Alg. 15(1), 123 (2005)

  5. 5.

    Klinger, A.: The Vandermonde matrix. Amer. Math. Monthly 74, 571–574 (1967)

  6. 6.

    Aceto, L., Trigiante, D.: The matrices of Pascal and other greats. Amer. Math. Monthly 108, 232–245 (2001)

  7. 7.

    Vein, R., Dale, P.: Determinants and their applications in mathematical physics. Springer-Verlag, New York, Inc. (1999). ISBN 0-387-98558-1

  8. 8.

    Yamaleev, R.M.: Geometrical and physical interpretation of evolution governed by general complex algebra. J. Math. Anal. Appl. 340, 1046–1057 (2008)

  9. 9.

    Cornelius Jr., E.F.: Identities for complete homogeneous symmetric polynomials. J. Algebr. Numb. Theor. Appl. 21(1), 109–116 (2011)

  10. 10.

    Blanchard, P., Devaney, R.L., Hall, G.R.: Differential Equations. Thompson. Coddington, E.A., Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill (2006)

  11. 11.

    Aceto, L., Trigiante, D.: The matrices of Pascal and classical polynomials. Rendicoti del Circol Matematico in Palermo. Serie II, Suppl. 68, 219–228 (2002)

  12. 12.

    Yamaleev, R.M.: Pascal matrix representation of evolution of polynmials. Int. J. Appl. Comput. Math. 1(4), 513–525 (2015)

  13. 13.

    Babusci, D., Dattoli, G., Di Palma, E., Sabia, E.: Adv. Appl. Cliff. Alg. 22(2), 271 (2012)

  14. 14.

    Kaufman, H.: A bibliografical note on the higher order sine functions. Scripta Math. 28, 28–36 (1967)

  15. 15.

    Yamaleev, R.M.: Difference between three quantities (2012). arXiv:1209.5012 [math.HO]

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Correspondence to Robert M. Yamaleev.

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Yamaleev, R.M. Divided Differences Calculus in Matrix Representation. Int. J. Appl. Comput. Math 5, 132 (2019).

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  • Vandermonde matrix
  • Divided differences
  • Trigonometry
  • Pascal matrix
  • Polynomial
  • Invariant functions