Some generalized rising binomial coefficients

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 13, 2007, Number 1, Pages 25—30
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Authors and affiliations

A. G. Shannon
Warrane College, The University of New South Wales, Kensington 1465, &
Raffles KvB, 99 Mount Street, North Sydney, NSW 2065, Australia

Abstract

This paper considers some properties of generalized binomial coefficients which are defined in terms of rising factorial coefficients. Analogies with classical results in number theory and some generalized special functions are highlighted.

Keywords

  • q-series
  • Binomial coefficients
  • Rising factorials
  • Generalized Fibonacci numbers
  • Appell set
  • Gauss—Cayley generalizations.

AMS Classification

  • 11B65
  • 11B39
  • 05A30

References

  1. Carlitz, L. Note on a Theorem of Glaisher. Journal of the London Mathematical Society. 28 (1953): 245-246.
  2. Carlitz, L. Generating Functions. The Fibonacci Quarterly. Vol.7 (1969), pp. 359-393.
  3. Jordan, C. Calculus of Finite Differences. New York: Chelsea.
  4. Macmahon, P.A. Combinatory Analysis. Cambridge: Cambridge University Press, 1916.
  5. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1958, p.3.
  6. Shannon, A.G. Some q-Binomial Coefficients Formed from Rising Factorials. Notes on Number Theory and Discrete Mathematics. 12 (2006): 13-20.
  7. Wright, E.M. An Identity and Applications. American Mathematical Monthly. Vol.75 (1968), pp.711-714.

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Cite this paper

APA

Shannon, A. G. (2007). Some generalized rising binomial coefficients. Notes on Number Theory and Discrete Mathematics, 13(1), 25-30.

Chicago

Shannon, AG. “Some Generalized Rising Binomial Coefficients.” Notes on Number Theory and Discrete Mathematics 13, no. 1 (2007): 25-30.

MLA

Shannon, AG. “Some Generalized Rising Binomial Coefficients.” Notes on Number Theory and Discrete Mathematics 13.1 (2007): 25-30. Print.

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