Gaussian binomial coefficients

A. G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 225-229
DOI: 10.7546/nntdm.2020.26.1.225-229
Full paper (PDF, 528 Kb)

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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 Gaussian binomial coefficients (and so-called) Fibonomial
coefficients) with identities related to Horadam’s generalized binomial coefficients.

Keywords

  • Gaussian binomial coefficients
  • Generalized Fibonacci numbers
  • Fibonomial coefficients

2010 Mathematics Subject Classification

  • 11B65
  • 11B39

References

  1. Alexanderson, G. L. (1974). A Fibonacci analogue of Gaussian binomial coefficients. Fibonacci Quarterly. 12 (2), 129–132.
  2. Andrews, G. E., & Baxter, R. J. (1987). Lattice gas generalization of the hard hexagon model. III. q-trinomial coefficients. Journal of Statistical Physics. 47 (3–4), 297–330.
  3. Andrews, G. E. (1970). A polynomial identity which implies the Rogers–Ramanujan identities. Scripta Mathematica. 28, 297–305.
  4. Andrews, G. E. (1974). Applications of basic hypergeometric functions. SIAM Review. 16, 441–484.
  5. Carlitz, L. (1947). A problem of Dickson’s. Duke Mathematical Journal. 14, 1139–1140.
  6. Carlitz, L. (1948). q-Bernoulli numbers and polynomials. Duke Mathematical Journal. 15, 987-1000.
  7. Carlitz, L. (1958). Expansions of q-Bernoulli numbers. Duke Mathematical Journal. 25, 355–364.
  8. Carlitz, L. (1961). Some integral equations satisfied by the complete integrals of first and second kind. Bolletino della Unione Matematica Italiana. (3) 16, 264–268.
  9. Carlitz, L. (1962). Generating functions for powers of certain sequences of numbers. Duke Mathematical Journal. 29, 521–537.
  10. Carlitz, L. (1968). A note on products of sequences. Bolletino della Unione Matematica Italiana. (4) 1, 362–365.
  11. Cross, J. T. (1983). The Euler φ-function in the Gaussian Integers. American Mathematical Monthly. 90 (8), 518–528.
  12. Gould, H. (1969). The bracket function and Fontene–Ward generalized binomial coefficients with application to Fibonomial coefficients. Fibonacci Quarterly. 7 (1), 23–40.
  13. Hoggatt, V. E, Jr, & Lind, D. A. (1968). Fibonacci and binomial properties of weighted compositions. Journal of Combinatorial Theory. 4, 121–124.
  14. Hoggatt, V. E. Jr. (1967). Fibonacci numbers and generalized binomial coefficients. The Fibonacci Quarterly. 5, 383–400.
  15. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly. 3, 161–176.
  16. Horadam, A. F. (1965). Generating functions for powers of a certain generalized sequence of numbers. Duke Mathematical Journal. 32, 437–446.
  17. Jerbic, S. K. (1968). Fibonomial Coefficients – A Few Summation Properties. San Jose State College, San Jose, California.
  18. Mercier, A. (1989). Identities containing Gaussian binomial coefficients. Discrete Mathematics. 76, 67–73.

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Cite this paper

Shannon, A. G.  (2020). Gaussian binomial coefficients. Notes on Number Theory and Discrete Mathematics, 26(1), 225-229, DOI: 10.7546/nntdm.2020.26.1.225-229.

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