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The Ramanujan Journal

, Volume 29, Issue 1–3, pp 51–67 | Cite as

Ramanujan-type partial theta identities and conjugate Bailey pairs

  • Jeremy Lovejoy
Article

Abstract

Residual identities of Ramanujan-type partial theta identities are tailor-made for producing conjugate Bailey pairs. This is carried out for partial theta identities in Ramanujan’s lost notebook, a number of the partial theta identities of Warnaar, and for some new ones as well.

Keywords

Bailey pairs Conjugate Bailey pairs Partial theta identities Mixed mock modular forms 

Mathematics Subject Classification (2000)

33D15 

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References

  1. 1.
    Andrews, G.E.: Ramanujan’s “lost” notebook, I: partial theta functions. Adv. Math. 41, 137–172 (1981) zbMATHCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The fifth and seventh order mock theta functions. Trans. Am. Math. Soc. 293, 113–134 (1986) zbMATHCrossRefGoogle Scholar
  3. 3.
    Andrews, G.E.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra. C.B.M.S. Regional Conference Series in Math, vol. 66. American Math. Soc., Providence (1986) Google Scholar
  4. 4.
    Andrews, G.E.: Bailey chains and generalized Lambert series: I. Four identities of Ramanujan. Ill. J. Math. 36, 251–274 (1992) zbMATHGoogle Scholar
  5. 5.
    Andrews, G.E., Dyson, F.J., Hickerson, D.: Partitions and indefinite quadratic forms. Invent. Math. 91, 391–407 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Andrews, G.E., Hickerson, D.: Ramanujan’s “lost” notebook, VII: The sixth order mock theta functions. Adv. Math. 89, 60–105 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Andrews, G.E., Warnaar, S.O.: The Bailey transform and false theta functions. Ramanujan J. 14, 173–188 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bailey, W.N.: Identities of the Rogers–Ramanujan type. Proc. Lond. Math. Soc. 50, 1–10 (1949) CrossRefGoogle Scholar
  9. 9.
    Berndt, B.C., Andrews, G.E.: Ramanujan’s Lost Notebook Part II. Springer, New York (2009) zbMATHGoogle Scholar
  10. 10.
    Bressoud, D.M.: Some identities for terminating q-series. Math. Proc. Camb. Philos. Soc. 89, 211–223 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Conley, C.H., Raum, M.: Harmonic Maass–Jacobi forms of degree 1 with higher rank indices. Preprint Google Scholar
  12. 12.
    Fine, N.: Basic Hypergeometric Series and Applications. American Mathematical Society, Providence (1988) zbMATHGoogle Scholar
  13. 13.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004) zbMATHCrossRefGoogle Scholar
  14. 14.
    Hickerson, D., Mortenson, E.: Hecke-type double sums, Appell–Lerch sums, and mock theta functions (I). Preprint Google Scholar
  15. 15.
    Hikami, K.: Hecke type formula for unified Witten–Reshetikhin–Turaev invariants as higher-order mock theta functions. Int. Math. Res. Not., rnm022 (2007) Google Scholar
  16. 16.
    Lovejoy, J.: Lacunary partition functions. Math. Res. Lett. 9, 191–198 (2002) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lovejoy, J.: A Bailey lattice. Proc. Am. Math. Soc. 132, 1507–1516 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Rowell, M.J.: A new general conjugate Bailey pair. Pac. J. Math. 238, 367–385 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Schilling, A., Warnaar, S.O.: Conjugate Bailey pairs: from configuration sums and fractional-level string functions to Bailey’s lemma. In: Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory (Charlottesville, VA, 2000). Contemp. Math., vol. 297, pp. 227–255. Amer. Math. Soc, Providence (2002) CrossRefGoogle Scholar
  20. 20.
    Singh, U.B.: A note on a transformation of Bailey. Q. J. Math. Oxf. Ser. (2) 45, 111–116 (1994) zbMATHCrossRefGoogle Scholar
  21. 21.
    Slater, L.J.: A new proof of Rogers’s transformations of infinite series. Proc. Lond. Math. Soc. 53, 460–475 (1951) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Warnaar, S.O.: Partial theta functions. I. Beyond the lost notebook. Proc. Lond. Math. Soc. 87, 363–395 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Zagier, D.: Ramanujan’s mock theta functions and their applications. Astérisque 326, 143–164 (2009) MathSciNetGoogle Scholar
  24. 24.
    Zwegers, S.: Mock theta functions. PhD Thesis, Utrecht University (2002) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.CNRS, LIAFAUniversité Denis Diderot-Paris 7Paris Cedex 13France

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