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Divided Differences Calculus in Matrix Representation

  • Robert M. YamaleevEmail author
Original Paper
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Abstract

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.

Keywords

Vandermonde matrix Divided differences Trigonometry Pascal matrix Polynomial Invariant functions 
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Notes

References

  1. 1.
    Dikoussar, N.D.: Method of basic elements. Math. Models Comput. Simulations 3(4), 492–507 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Carl, De Boor: Divided Differenes. Surv. Approx. Theory 1, 46–69 (2005)MathSciNetGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Klinger, A.: The Vandermonde matrix. Amer. Math. Monthly 74, 571–574 (1967)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Aceto, L., Trigiante, D.: The matrices of Pascal and other greats. Amer. Math. Monthly 108, 232–245 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Vein, R., Dale, P.: Determinants and their applications in mathematical physics. Springer-Verlag, New York, Inc. (1999). ISBN 0-387-98558-1zbMATHGoogle Scholar
  8. 8.
    Yamaleev, R.M.: Geometrical and physical interpretation of evolution governed by general complex algebra. J. Math. Anal. Appl. 340, 1046–1057 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cornelius Jr., E.F.: Identities for complete homogeneous symmetric polynomials. J. Algebr. Numb. Theor. Appl. 21(1), 109–116 (2011)MathSciNetzbMATHGoogle Scholar
  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)Google Scholar
  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)Google Scholar
  12. 12.
    Yamaleev, R.M.: Pascal matrix representation of evolution of polynmials. Int. J. Appl. Comput. Math. 1(4), 513–525 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Babusci, D., Dattoli, G., Di Palma, E., Sabia, E.: Adv. Appl. Cliff. Alg. 22(2), 271 (2012)CrossRefGoogle Scholar
  14. 14.
    Kaufman, H.: A bibliografical note on the higher order sine functions. Scripta Math. 28, 28–36 (1967)Google Scholar
  15. 15.
    Yamaleev, R.M.: Difference between three quantities (2012). arXiv:1209.5012 [math.HO]

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© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Joint Institute for Nuclear ResearchLITDubnaRussia

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