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The Ramanujan Journal

, Volume 47, Issue 2, pp 309–316 | Cite as

Special classes of q-bracket operators

  • Tanay Wakhare
Article

Abstract

We study the q-bracket operator of Bloch and Okounkov, recently examined by Zagier and other authors, when applied to functions defined by two classes of sums over the parts of an integer partition. We derive convolution identities for these functions and link both classes of q-brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s Theorem on the number of ones in all partitions of n, and provide several new combinatorial results.

Keywords

Partitions q-Series Multiplicative number theory Additive number theory 

Mathematics Subject Classification

05A17 11P81 11N99 

Notes

Acknowledgements

I would like to thank Christophe Vignat for guiding me through my first forays into mathematics, and Armin Straub for first introducing me to Stanley’s Theorem and the current literature on partitions. I would also like to thank George Andrews, Robert Schneider, and the anonymous referee for helpful comments. Last but not least, I would like to thank Ellicott 3 for always providing me with (questionable) inspiration.

References

  1. 1.
    Alladi, K., Erdõs, P.: On an additive arithmetic function. Pac. J. Math. 71, 275–294 (1977)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing Co., Reading (1976)zbMATHGoogle Scholar
  3. 3.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)zbMATHGoogle Scholar
  4. 4.
    Bloch, S., Okounkov, A.: The character of the infinite wedge representation. Adv. Math. 149, 1–60 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Grifffin, M.J., Jameson, M., Trebat-Leder, S.: On \(p\)-adic modular forms and the Bloch–Okounkov theorem. Res. Math. Sci. 3(11), 14 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Han, G.N.: An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, Preprint (2008)Google Scholar
  7. 7.
    Hirschhorn, M.D.: The number of different parts in the partitions of \(n\). Fibonacci Q. 52, 10–15 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov, Release 1.0.9 of 2014-08-29. Online companion to [8]
  9. 9.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York. Print companion to [9] (2010)Google Scholar
  10. 10.
    Schneider, R.: Arithmetic of partitions and the \(q\)-bracket operator. Proc. Am. Math. Soc. 145, 1953–1968 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1, 2nd edn. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  12. 12.
    Zagier, D.: Partitions, quasimodular forms, and the Bloch–Okounkov theorem. Ramanujan J. 41, 1–24 (2015)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.University of Maryland, College ParkCollege ParkUSA

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