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Abel's Convolution Formulae through Taylor Polynomials

Year 2023, Volume: 5 Issue: 2, 47 - 51, 30.11.2023
https://doi.org/10.47087/mjm.1314434

Abstract

By making use of the Taylor polynomials, new proofs are presented
for three binomial identities including Abel’s convolution formula.

References

  • N. H. Abel, Beweis eines Ausdrucks, von welchem die Binomial–Formel ein einzelner Fall ist, J. Reine Angew. Math. 1 (1826), 159–160.
  • W. Chu, Inversion techniques and combinatorial identities: A quick introduction to hypergeometric evaluations, Math. Appl. 283 (1994), 31–57.
  • W. Chu, Generating functions and combinatorial identities, Glas. Mat. 33 (1998), 1–12.
  • W. Chu, Elementary Proofs for Convolution Identities of Abel and Hagen–Rothe, Electron. J. Combin. 17 (2010), N24.
  • W. Chu, Finite differences and terminating hypergeometric series, Bull. Irish Math. Soc. 78 (2016), 31–45.
  • W. Chu and L. C. Hsu, Some new applications of Gould-Hsu inversions, J. Combin. Inf. Syst. Sci. 14:1 (1990), 1–4.
  • L. Comtet, Advanced Combinatorics, Dordrecht–Holland, The Netherlands, 1974.
  • G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Translated from the Russian by H. H. McFadden: Translations of Mathematical Monographs 59; American Mathematical Society, Providence, RI, 1984. 286pp.
  • H. W. Gould, Some generalizations of Vandermonde’s convolution, Amer. Math. Monthly 63:1 (1956), 84–91.
  • H. W. Gould, Generalization of a theorem of Jensen concerning convolutions, Duke Math. J. 27 (1960), 71–76.
  • H. W. Gould, Combinatorial Identities: a standardized set of tables listing 500 binomial coefficient summations, West Virginia University, Morgantown, 1972.
  • H. W. Gould and L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics (2nd edition), AddisonWesley Publ. Company, Reading, Massachusetts, 1994.
  • J. L. W. V. Jensen, Sur une identite d Abel et sur d’autres formules analogues, Acta Math. 26 (1902), 307–318.
  • S. G. Mohanty, Lattice Path Counting and Applications, Z. W. Birnbaum and E. Lukacs, 1979.
  • T. V. Narayana, Lattice Path Combinatorics with Statistical Applications, University of Toronto Press, Toronto - 1979.
  • J. Riordan, Combinatorial Identities, John Wiley Sons, Inc. New York - 1968.
  • R. Sprugnoli, Riordan arrays and the Abel–Gould identity, Discrete Math. 142 (1995), 213– 233.
  • V. Strehl, Identities of Rothe–Abel–Schl¨afli–Hurwitz–type, Discrete Math. 99:1–3 (1992), 321– 340.
  • P. J. Zucker, Problem 1578, Math. Mag. 72:1 (1999), page 237; Solution: ibid 73:3 (2000), 243–245.
Year 2023, Volume: 5 Issue: 2, 47 - 51, 30.11.2023
https://doi.org/10.47087/mjm.1314434

Abstract

References

  • N. H. Abel, Beweis eines Ausdrucks, von welchem die Binomial–Formel ein einzelner Fall ist, J. Reine Angew. Math. 1 (1826), 159–160.
  • W. Chu, Inversion techniques and combinatorial identities: A quick introduction to hypergeometric evaluations, Math. Appl. 283 (1994), 31–57.
  • W. Chu, Generating functions and combinatorial identities, Glas. Mat. 33 (1998), 1–12.
  • W. Chu, Elementary Proofs for Convolution Identities of Abel and Hagen–Rothe, Electron. J. Combin. 17 (2010), N24.
  • W. Chu, Finite differences and terminating hypergeometric series, Bull. Irish Math. Soc. 78 (2016), 31–45.
  • W. Chu and L. C. Hsu, Some new applications of Gould-Hsu inversions, J. Combin. Inf. Syst. Sci. 14:1 (1990), 1–4.
  • L. Comtet, Advanced Combinatorics, Dordrecht–Holland, The Netherlands, 1974.
  • G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Translated from the Russian by H. H. McFadden: Translations of Mathematical Monographs 59; American Mathematical Society, Providence, RI, 1984. 286pp.
  • H. W. Gould, Some generalizations of Vandermonde’s convolution, Amer. Math. Monthly 63:1 (1956), 84–91.
  • H. W. Gould, Generalization of a theorem of Jensen concerning convolutions, Duke Math. J. 27 (1960), 71–76.
  • H. W. Gould, Combinatorial Identities: a standardized set of tables listing 500 binomial coefficient summations, West Virginia University, Morgantown, 1972.
  • H. W. Gould and L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics (2nd edition), AddisonWesley Publ. Company, Reading, Massachusetts, 1994.
  • J. L. W. V. Jensen, Sur une identite d Abel et sur d’autres formules analogues, Acta Math. 26 (1902), 307–318.
  • S. G. Mohanty, Lattice Path Counting and Applications, Z. W. Birnbaum and E. Lukacs, 1979.
  • T. V. Narayana, Lattice Path Combinatorics with Statistical Applications, University of Toronto Press, Toronto - 1979.
  • J. Riordan, Combinatorial Identities, John Wiley Sons, Inc. New York - 1968.
  • R. Sprugnoli, Riordan arrays and the Abel–Gould identity, Discrete Math. 142 (1995), 213– 233.
  • V. Strehl, Identities of Rothe–Abel–Schl¨afli–Hurwitz–type, Discrete Math. 99:1–3 (1992), 321– 340.
  • P. J. Zucker, Problem 1578, Math. Mag. 72:1 (1999), page 237; Solution: ibid 73:3 (2000), 243–245.
There are 20 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Articles
Authors

Wenchang Chu 0000-0002-8425-212X

Early Pub Date November 30, 2023
Publication Date November 30, 2023
Acceptance Date November 24, 2023
Published in Issue Year 2023 Volume: 5 Issue: 2

Cite

APA Chu, W. (2023). Abel’s Convolution Formulae through Taylor Polynomials. Maltepe Journal of Mathematics, 5(2), 47-51. https://doi.org/10.47087/mjm.1314434
AMA Chu W. Abel’s Convolution Formulae through Taylor Polynomials. Maltepe Journal of Mathematics. November 2023;5(2):47-51. doi:10.47087/mjm.1314434
Chicago Chu, Wenchang. “Abel’s Convolution Formulae through Taylor Polynomials”. Maltepe Journal of Mathematics 5, no. 2 (November 2023): 47-51. https://doi.org/10.47087/mjm.1314434.
EndNote Chu W (November 1, 2023) Abel’s Convolution Formulae through Taylor Polynomials. Maltepe Journal of Mathematics 5 2 47–51.
IEEE W. Chu, “Abel’s Convolution Formulae through Taylor Polynomials”, Maltepe Journal of Mathematics, vol. 5, no. 2, pp. 47–51, 2023, doi: 10.47087/mjm.1314434.
ISNAD Chu, Wenchang. “Abel’s Convolution Formulae through Taylor Polynomials”. Maltepe Journal of Mathematics 5/2 (November 2023), 47-51. https://doi.org/10.47087/mjm.1314434.
JAMA Chu W. Abel’s Convolution Formulae through Taylor Polynomials. Maltepe Journal of Mathematics. 2023;5:47–51.
MLA Chu, Wenchang. “Abel’s Convolution Formulae through Taylor Polynomials”. Maltepe Journal of Mathematics, vol. 5, no. 2, 2023, pp. 47-51, doi:10.47087/mjm.1314434.
Vancouver Chu W. Abel’s Convolution Formulae through Taylor Polynomials. Maltepe Journal of Mathematics. 2023;5(2):47-51.

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