Domain Extensions of Binomial Numbers Applying Successive Sums Transformations on Sequences Indexed by Integers

Authors

  • Marlo M. Barroso National Laboratory of Scientific Computation, LNCC, Petrópolis, RJ, Av. Getulio Vargas, 333 - Quitandinha, 25651-075, Petrópolis, RJ, Brasil. https://orcid.org/0000-0002-6093-3828
  • José Karam-Filho National Laboratory of Scientific Computation, LNCC, Petrópolis, RJ, Av. Getulio Vargas, 333 - Quitandinha, 25651-075, Petrópolis, RJ, Brasil.
  • Gilson A. Giraldi National Laboratory of Scientific Computation, LNCC, Petrópolis, RJ, Av. Getulio Vargas, 333 - Quitandinha, 25651-075, Petrópolis, RJ, Brasil.

DOI:

https://doi.org/10.5540/tema.2020.021.01.133

Keywords:

Discrete Mathematics, Algebraic Structures, Recursion Sequences, Successive Product, Successive Sum, Binomial Numbers,

Abstract

The classic definition of binomial numbers involves factorials, making unfeasible their extension for negative integers. The methodology applied in this paper allows to establish several new binomial numbers extensions for the integer domain, reproduces to integer arguments those extensions that are proposed in other works, and also verifies the results of the usual binomial numbers. To do this, the impossibility to compute factorials with negative integer arguments is eliminated by the replacement of the classic binomial definition to a new one, based on operations recently proposed and, until now, referred to as transformations by the successive sum applied on sequences indexed by integers. By particularizing these operations for the sequences formed and indexed by integers, it is possible to redefine the usual binomial numbers to any integer arguments, with the advantage that the values are more easily computed by using successive additions instead of multiplications, divisions or possibly more elaborate combinations of these operators, which could demand more than one or two sentences to their application.

Author Biographies

Marlo M. Barroso, National Laboratory of Scientific Computation, LNCC, Petrópolis, RJ, Av. Getulio Vargas, 333 - Quitandinha, 25651-075, Petrópolis, RJ, Brasil.

Department of Mathematical and Computing Methods - Consultant, since 2017.

José Karam-Filho, National Laboratory of Scientific Computation, LNCC, Petrópolis, RJ, Av. Getulio Vargas, 333 - Quitandinha, 25651-075, Petrópolis, RJ, Brasil.

Department of Computing Modeling - National Laboratory
of Scientic Computation - Research Associate, since 1988

Gilson A. Giraldi, National Laboratory of Scientific Computation, LNCC, Petrópolis, RJ, Av. Getulio Vargas, 333 - Quitandinha, 25651-075, Petrópolis, RJ, Brasil.

Department of Mathematical and Computing Methods - Research Associate, since 2002

References

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M. Moesia B., J. Karam F., and G. A. Giraldi, "Uma nova metodologia para a extensão de domínio de operações matemáticas sucessivas," in Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, vol. 6, n.2, SBMAC,

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Published

2020-03-27

How to Cite

Barroso, M. M., Karam-Filho, J., & Giraldi, G. A. (2020). Domain Extensions of Binomial Numbers Applying Successive Sums Transformations on Sequences Indexed by Integers. Trends in Computational and Applied Mathematics, 21(1), 133. https://doi.org/10.5540/tema.2020.021.01.133

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Section

Original Article