Skip to main content

Combinatorial Interpretation of a Generalized Basic Series

  • 2034 Accesses

Abstract

Recently Goyal and Agarwal (ARS Combinatoria, to appear) have interpreted a generalized basic series as a generating function for a colour partition function and a weighted lattice path function. This resulted in an infinite family of combinatorial identities. Using a bijection between the Bender–Knuth matrices and the n-colour partitions established by the first author in Agarwal (ARS Combinatoria, 61, 97–117, 2001), in this paper we extend the main result of Goyal and Agarwal to a 3-way infinite family of combinatorial identities. We illustrate by two examples that our main result has the potential of yielding many Rogers–Ramanujan–MacMahon type combinatorial identities.

Keywords

  • Basal Serum
  • Combinatorial Identities
  • Infinite Family
  • Main Result
  • Bijection

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-1-4939-0258-3_7
  • Chapter length: 11 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-1-4939-0258-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.99
Price excludes VAT (USA)
Hardcover Book
USD   199.99
Price excludes VAT (USA)
Fig. 1

References

  1. Agarwal, A.K.: Rogers-Ramanujan identities for n-color partitions. J. Number Theory 28, 299–305 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Agarwal, A.K.: New combinatorial interpretations of two analytic identities. Proc. Am. Math. Soc. 107(2), 561–567 (1989)

    CrossRef  MATH  Google Scholar 

  3. Agarwal, A.K.: q-functional equations and some partition identities, combinatorics and theoretical computer science (Washington, 1989). Discrete Appl. Math. 34(1–3), 17–26 (1991)

    Google Scholar 

  4. Agarwal, A.K.: New classes of infinite 3-way partition identities. ARS Combinatoria 44, 33–54 (1996)

    MATH  MathSciNet  Google Scholar 

  5. Agarwal, A.K.: n- color analogues of Gaussian polynomials. ARS Combinatoria 61, 97–117 (2001)

    MathSciNet  Google Scholar 

  6. Agarwal, A.K., Andrews, G.E.: Hook differences and lattice paths. J. Statist. Plann. Inference 14(1), 5–14 (1986)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Agarwal, A.K., Andrews, G.E.: Rogers-Ramanujan identities for partitions with “N copies of N”. J. Combin. Theory Ser. A. 45(1), 40–49 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Agarwal, A.K., Bressoud, D.M.: Lattice paths and multiple basic hypergeometric series. Pacific J. Math. 136(2), 209–228 (1989)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Alladi, K., Berkovich, A.: Gollnitz-Gordon partitions with weights and parity conditions. In: Aoki, T., Kanemitsu, S., Nakahara, M., Ohno, Y. (eds.) Zeta Functions, Topology and Quantum Physics. Developments in Mathematics, vol. 14, pp. 1–17. Springer, New York (2005)

    CrossRef  Google Scholar 

  10. Andrews, G.E.: An introduction to Ramanujan’s “Lost” notebook. Am. Math. Monthly 86, 89–108 (1979)

    CrossRef  MATH  Google Scholar 

  11. Bender, E.A., Knuth, D.E.: Enumeration of plane partitions. J. Combin. Theory (A) 13, 40–54 (1972)

    Google Scholar 

  12. Conner, W.G.: Partition theorems related to some identities of Rogers and Watson. Trans. Am. Math. Soc. 214, 95–111 (1975)

    CrossRef  Google Scholar 

  13. Göllnitz, H.: Einfache partitionen (unpublished). Diplomarbeit W.S., 65 pp., Gotttingen (1960)

    Google Scholar 

  14. Göllnitz, H.: Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. 225, 154–190 (1967)

    MATH  MathSciNet  Google Scholar 

  15. Gordon, B.: Some continued fractions of the Rogers-Ramanujan type. Duke J. Math. 32, 741–748 (1965)

    CrossRef  MATH  Google Scholar 

  16. Goyal, M., Agarwal, A.K.: Further Rogers-Ramanujan identities for n-color partitions. Utilitas Mathematica (to appear)

    Google Scholar 

  17. Goyal, M., Agarwal, A.K.: On a new class of combinatorial identities. ARS Combinatoria (to appear)

    Google Scholar 

  18. Hirschhorm, M.D.: Some partition theorems of the Rogers-Ramanujan type. J. Combin. Theory, Ser. A 27(1), 33–37 (1979)

    Google Scholar 

  19. MacMahan, P.A.: Combinatory Analysis, vol. 2. Cambridge University Press, London and Newyork (1916)

    Google Scholar 

  20. Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)

    Google Scholar 

  21. Slater, L.J.: Further identities of the Rogers-Ramanujan type. Proc. London Math. Soc. 54, 147–167 (1952)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Subbarao, M.V.: Some Rogers-Ramanujan type partition theorems. Pacific J. Math. 120, 431–435 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Subbarao, M.V., Agarwal, A.K.: Further theorems of the Rogers-Ramanujan type. Canad. Math. Bull. 31(2), 210–214 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author is an emeritus scientist of the Council of Scientific and Industrial Research (CSIR), Government of India. He was supported by CSIR Research Scheme No. 21(0879)/11/EMR-II.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. K. Agarwal .

Editor information

Editors and Affiliations

Additional information

Dedicated to Professor Hari M. Srivastava

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Agarwal, A.K., Rana, M. (2014). Combinatorial Interpretation of a Generalized Basic Series. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_7

Download citation