A q-series Bernoulli—Euler partition formula

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 14, 2008, Number 1, Pages 6—9
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Authors and affiliations

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

Absrtacts

This paper utilized a modification of q-series to develop some partition formulas in the tradition of Bernoulli—Euler—Rogers—Ramanujan identities. Many of the ideas owe their development to the detailed pioneering work of Leonard Carlitz, with the added acknowledgement to the creative work currently being done by other number theorists working in this fertile area.

AMS Classification

  • 11B65
  • 11B39
  • 05A30

References

  1. Andrews, G.E. A Polynomial Identity Which Implies the Rogers-Ramanujan Identities. Scripta Mathematica. 28 (1970): 297-305.
  2. Carlitz, L. q-Bernoulli Numbers and Polynomials. Duke Mathematical Journal. 15 (1948): 987-1000.
  3. Carlitz, L. Expansions of q-Bernoulli Numbers. Duke Mathematical Journal. 25 (1958): 355-364.
  4. Carlitz, L. A Note on the Rogers-Ramanujan Identities. Mathematische Nachrichten. 17 (1958): 23-26.
  5. Carlitz, L. Some Formulas Related to the Rogers-Ramanujan Identities. Annali di Matematica Pura ed Applicata. (4) 47 (1959): 243-251.
  6. 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.
  7. Carlitz, L. Characterization of Certain Sequences of Orthogonal Polynomials. Portugaliae Mathematica. 20 (1961): 43-46.
  8. Carlitz, L. Generating Functions for Powers of Certain Sequences of Numbers. Duke Mathematical Journal. 29 (1962): 521-537.
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  11. Carlitz, L. Some Partition Problems Related to the Stirling Numbers of the Second Kind. Acta Arithmetica. 10 (1965): 409-422.
  12. Carlitz, L. Note on Some Continued Fractions of the Rogers-Ramanujan Type. Duke Mathematical Journal. 32 (1965): 713-720.
  13. Carlitz, L. Rectangular Arrays and Plane Partitions. Acta Arithmetica. 13 (1967): 29-47.
  14. Carlitz, L. A Note on Products of Sequences. Bolletino della Unione Matematica Italiana. (4) 1 (1968): 362-365.
  15. Carlitz, L. A Note on the Rogers-Ramanujan Identities. Duke Mathematical Journal. 35 (1968): 839-842.Carlitz, L. Fibonacci Representations. II. The Fibonacci Quarterly. 8 (1970): 113-134.
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  17. Ernst, T. Examples of a q-umbral Calculus. Advanced Studies in Contemporary Mathematics. 16 (2008): 1-22.
  18. Kim, T. Some Formulae for the q-Bernoulli and Euler Polynomials. Journal of Mathematical Analysis and Applications. 273 (2002): 236-242.
  19. Kim, T. Analytic Continuation of Multiple q-zeta Functions and their Values at Negative Integers. Russian Journal of Mathematics & Physics. 11 (2004): 71-76.
  20. Kim, T. & S.H. Rim. On Changed q-Euler Numbers and Polynomials. Advanced Studies in Contemporary Mathematics. 9 (2004): 81-86.
  21. Sylvester, J.J. The Collected Mathematical Papers of James Joseph Sylvester. Volume IV (1882-1897). New York: Chelsea, 1973, pp.91,93-94.

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Cite this paper

APA

Shannon A. G. (2008). A q-series Bernoulli—Euler partition formula. Notes on Number Theory and Discrete Mathematics, 14(1), 6-9.

Chicago

Shannon, AG. “A q-series Bernoulli—Euler Partition Formula.” Notes on Number Theory and Discrete Mathematics 14, no. 1 (2008): 6-9.

MLA

Shannon, AG. “A q-series Bernoulli—Euler Partition Formula.” Notes on Number Theory and Discrete Mathematics 14.1 (2008): 6-9. Print.

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