A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 12, 2006, Number 1, Pages 13–20

**Full paper (PDF, 102 Kb)**

## Details

### Authors and affiliations

A. G. Shannon

*Warrane College, The University of New South Wales, Kensington 1465, &
KvB Institute of Technology, 99 Mount Street, North Sydney, NSW 2065, Australia
*

### Abstract

This paper considers some *q*-extensions of binomial coefficients. Some of the results are applied to some generalized Fibonacci numbers, and others are included as ideas for further investigation, particularly, particularly into *q*-Bernoulli polynomials.

### AMS Classification

- 11B65
- 11B39
- 05A30

### References

- G.E. Andrews. A Polynomial Identity Which Implies the Rogers-Ramanujan Identities. Scripta Mathematica. 28 (1970): 297-305.
- Sacha C. Blumen, Two Generalizations of the Binomial Theorem, The Australian Mathematical Society Gazette, 33 (2006): 39-43.
- L. Carlitz. q-Bernoulli Numbers and Polynomials. Duke Mathematical Journal. 15 (1948): 987-1000.
- L. Carlitz. Expansions of q-Bernoulli Numbers. Duke Mathematical Journal. 25 (1958): 355-364.
- L. Carlitz. Some Integral Equations satisfied by the Complete Elliptic Integrals of the First and Second Kind. Bolletino della Unione Matematica Italiana. (3) 16 (1961): 264-268.
- L. Carlitz. Characterization of Certain Sequences of Orthogonal Polynomials. Portugaliae Mathematica. 20 (1961): 43-46.
- L. Carlitz. Generating Functions for Powers of Certain Sequences of Numbers. Duke Mathematical Journal. 29 (1962): 521-537.
- L. Carlitz. A q-identity. Monatshefte für Mathematik. 67 (1963): 305-310.
- L. Carlitz. A Note on Products of Sequences. Bolletino della Unione Matematica Italiana. (4) 1 (1968): 362-365.
- B. Gordon and L. Houten. Notes on Plane Partitions. I. Journal of Combinatorial Theory. 4 (1968): 72-80.
- B. Gordon and L. Houten. Notes on Plane Partitions. II. Journal of Combinatorial Theory. 4 (1968): 81-99.
- V.E. Hoggatt, Jr. Fibonacci Numbers and Generalized Binomial Coefficients. The Fibonacci Quarterly. 5 (1967): 383-400.
- A.F. Horadam. Generating Functions for Powers of a Certain Generalized Sequence of Numbers. Duke Mathematical Journal. 32 (1965): 437-446.
- S.K. Jerbic. Fibonomial Coefficients – A Few Summation Properties. Master’s Thesis, San José State College, California, 1968.
- T. Kim, Some Formulae for the q-Bernoulli and Euler Polynomials. Journal of Mathematical Analysis and Applications. 273 (2002): 236-242.
- T. Kim. Analytic Continuation of Multiple q-zeta Functions and their Values at Negative Integers. Russian Journal of Mathematics & Physics. 11 (2004): 71-76.
- T. Kim and S.H. Rim. On Changed q-Euler Numbers and Polynomials. Advanced Studies in Contemporary Mathematics. 9 (2004): 81-86.
- P.A. Macmahon. Combinatory Analysis. Cambridge: Cambridge University Press, 1916.
- R.L. Ollerton and A.G. Shannon. Extensions of Generalized Binomial Coefficients. In F. Howard (ed.), Applications of Fibonacci Numbers, Volume 9. Dordrecht: Kluwer, 2004, pp.187-199.
- J. Riordan. An Introduction to Combinatorial Analysis. New York: Wiley, 1958, p.3.

## Related papers

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

## Cite this paper

Shannon, A. G. (2006). Some *q*-binomial coefficients. *Notes on Number Theory and Discrete Mathematics*, 12(1), 13-20.