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Divided Differences Calculus in Matrix Representation

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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.

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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). https://doi.org/10.1007/s40819-019-0719-7

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Keywords

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