Some q-series inversion formulae

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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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

  1. Carlitz, L. q-Bernoulli Numbers and Polynomials. Duke Mathematical Journal. 15 (1948): 987-1000.
  2. Carlitz, L. Expansions of q-Bernoulli Numbers. Duke Mathematical Journal. 25 (1958): 355-364.
  3. 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.
  4. Carlitz, L. Characterization of Certain Sequences of Orthogonal Polynomials. Portugaliae Mathematica. 20 (1961): 43-46.
  5. Carlitz, L. Generating Functions for Powers of Certain Sequences of Numbers. Duke Mathematical Journal. 29 (1962): 521-537.
  6. Carlitz, L. A q-identity. Monatshefte fur Mathematik. 67 (1963): 305-310.
  7. Carlitz, L. A Note on Products of Sequences. Bolletino della Unione Matematica Italiana. (4) 1 (1968): 362-365.
  8. Carlitz, L. Generating Functions. The Fibonacci Quarterly. 7 (1969): 359-393.
  9. Cauchy, A.L. Memoire sur les functions don plusiers valuers. Comptes Rendus de l’Academie des Sciences. 17 (1843): 526-534.
  10. Kim, T. Some Formulae for the q-Bernoulli and Euler Polynomials. Journal of Mathematical Analysis and Applications. 273 (2002): 236-242.
  11. Kim, T. Analytic Continuation of Multiple q-zeta Functions and their Values at Negative Integers. Russian Journal of Mathematics & Physics. 11 (2004): 71-76.
  12. 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

APA

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

Chicago

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

MLA

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

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