A. G. Shannon

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

Volume 13, 2007, Number 2, Pages 15—18

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## Details

### 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 *q*-extensions of binomial coefficients formed from rising factorial coefficients. Some of the results are applied to a Möbius Inversion Formula based on extensions of ideas initially developed by Leonard Carlitz.

### Keywords

*q*-series- Binomial coefficients
- Möbius function
- Rising factorials
- Exponential functions

### AMS Classification

- 11B65
- 11B39
- 05A30

### References

- Carlitz, L.
*q*-Bernoulli Numbers and Polynomials. Duke Mathematical Journal. 15 (1948): 987-1000. - Carlitz, L. Expansions of
*q*-Bernoulli Numbers. Duke Mathematical Journal. 25 (1958): 355-364. - Carlitz L. 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.
- Carlitz, L. Characterization of Certain Sequences of Orthogonal Polynomials. Portugaliae Mathematica. 20 (1961): 43-46.
- Carlitz, L. Generating Functions for Powers of Certain Sequences of Numbers. Duke Mathematical Journal. 29 (1962): 521-537.
- Carlitz, L. A
*q*-identity. Monatshefte fur Mathematik. 67 (1963): 305-310. - Carlitz, L. A Note on Products of Sequences. Bolletino della Unione Matematica Italiana. (4) 1 (1968): 362-365.
- Carlitz, L. Generating Functions. The Fibonacci Quarterly. 7 (1969): 359-393.
- Cauchy, A.L. Memoire sur les functions don plusiers valuers. Comptes Rendus de l’Academie des Sciences. 17 (1843): 526-534.
- Kim, T. Some Formulae for the
*q*-Bernoulli and Euler Polynomials. Journal of Mathematical Analysis and Applications. 273 (2002): 236-242. - Kim, T. Analytic Continuation of Multiple
*q*-zeta Functions and their Values at Negative Integers. Russian Journal of Mathematics & Physics. 11 (2004): 71-76. - Kim, T. & S.H. Rim. On Changed
*q*-Euler Numbers and Polynomials. Advanced Studies in Contemporary Mathematics. 9 (2004): 81-86.

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## Cite this paper

APAShannon, A. G. (2007). Some *q*-series inversion formulae. Notes on Number Theory and Discrete Mathematics, 13(2), 15-18.

Shannon, AG. “Some *q*-series Inversion Formulae.” Notes on Number Theory and Discrete Mathematics 13, no. 2 (2007): 15-18.

Shannon, AG. Some *q*-series Inversion Formulae.” Notes on Number Theory and Discrete Mathematics 13.2 (2007): 15-18. Print.