Multisection of series

A. G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 4, Pages 66—71
DOI: 10.7546/nntdm.2019.25.4.66-71
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Authors and affiliations

A. G. Shannon
Warrane College, The University of New South Wales,
Kensington, NSW 2033, Australia

Abstract

This paper extends some ideas of Leonard Carlitz with ideas from John Riordan and
validated against results of Alwyn Horadam. They involve breaking up series and considering multiple sums of ordinary and generalized binomial coefficients.

Keywords

  • Generating function
  • Binomial coefficient
  • Primitive root
  • Lucas numbers
  • Kronecker delta
  • Binet formulas
  • q-series

2010 Mathematics Subject Classification

  • 11B65
  • 11B39

References

  1. Carlitz, L. (1965). Multiple sums and generating functions. Collectanea Mathematica. 17, 281–296.
  2. Riordan, J. (1958). An Introduction to Combinatorial Analysis. New York: Wiley, p.3.
  3. Carlitz, L. (1967). Rectangular arrays and plane partitions. Acta Arithmetica. 13, 29–47.
  4. Carlitz, L. (1970). Some generalized Fibonacci identities. The Fibonacci Quarterly, 8, 249–254.
  5. Carlitz, L. (1965). Some multiple sums and binomial identities. SIAM Journal on Applied Mathematics, 13, 469–486.
  6. Carlitz, L. (1964). A binomial identity arising from a sorting problem. SIAM Review, 6, 20–30.
  7. Carlitz, L. (1967). A binomial identity. SIAM Review, 229–231.
  8. Shannon, A. G. (1980). Some lacunary recurrence relations. The Fibonacci Quarterly. 18, 73–79.
  9. Aistleitner, C., Berkes , I., & Tichy, R. (2010). Lacunary sequences and permutations. In I. Berkes, R. Bradley, M. Peligrad and R. Tichy. Dependence in Probability, Analysis and Number Theory: A Volume in Memory of Walter Philipp. Heber City, UT: Kendrick Press, 35–49.
  10. Lehmer, D. H. (1935). Lacunary recurrence formulas for the numbers of Bernoulli and Euler. Annals of Mathematics, 36, 637–649.
  11. Benjamin, A. T., Chen, B., & Kindred, K. (2010). Sums of evenly spaced binomial
    coefficients. Mathematics Magazine, 83, 310–373.
  12. Beauregard, R. A., & Dobrushkin, V. A. (2016). Multisection of series. The Mathematical Gazette. 100, 549, 460–470.
  13. Horadam, A. F. (1965). Generating functions for powers of a certain generalized sequence of numbers. Duke Mathematical Journal, 32, 437–446.
  14. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3, 161–176.
  15. Carlitz, L., & Riordan, J. (1964). Two element lattice permutation numbers and their q-generalization. Duke Mathematical Journal, 31, 317–388.
  16. Carlitz, L. (1962). Note on a q-identity. Univerzitet u Beogradu. Publikacije
    Elektrotehničkeg Fakulteta. Serija Matematika i Fizika, 78–83, 19–20.
  17. Carlitz, L. (1955). Note on a q-identity. Mathematica Scandinavica, 3, 281–282.
  18. Shannon, A. G. (2014). Fermatian Hurwitz Series within the Fontené–Jackson Calculus. Advanced Studies in Contemporary Mathematics, 24 (1), 129–136.
  19. Heine, E. 1847. Untersuchungen über die Reihe 1 + … x2 + … . Journal für die Reine und Angewandte Mathematik, 34, 285–328.
  20. Riordan, J. (1968). Combinatorial Identities. New York: Wiley, pp. 45, 202.
  21. Carlitz, L. (1954). Note on Legendre polynomials. Bulletin of the Calcutta Mathematical Society, 46, 93–95.
  22. Shannon, A. G. (2003). Generalized Bernoulli Polynomials and Jackson’s Calculus of Sequences. Notes on Number Theory and Discrete Mathematics, 9 (1), 1–6.
  23. Horadam, A. F., & Shannon, A. G. (1976). Ward’s Staudt-Clausen problem. Mathematica Scandinavica, 29, 239–250.
  24. Gould, H. W. (1961). The q-series generalization of a formula of Sparre Andersen.
    Mathematica Scandinavica, 9, 90–94.
  25. Gould, H. W. (1958). Note on a paper of Sparre Andersen. Mathematica Scandinavica, 6, 226–230.

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Cite this paper

APA

Shannon, A. G. (2019). Multisection of series. Notes on Number Theory and Discrete Mathematics, 25(4), 66-71, doi: 10.7546/nntdm.2019.25.4.66-71.

Chicago

Shannon, A. G. “Multisection of Series.” Notes on Number Theory and Discrete Mathematics 25, no. 4 (2019): 66-71, doi: 10.7546/nntdm.2019.25.4.66-71.

MLA

Shannon, A. G. “Multisection of Series.” Notes on Number Theory and Discrete Mathematics 25.4 (2019): 66-71. Print, doi: 10.7546/nntdm.2019.25.4.66-71.

 

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