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Bailey pairs and indefinite quadratic forms, II. False indefinite theta functions


We construct families of Bailey pairs \((\alpha _n,\beta _n)\) where the exponent of q in \(\alpha _n\) is an indefinite quadratic form, but where the usual \((-1)^j\) is replaced by a sign function. This leads to identities involving “false” indefinite binary theta series. These closely resemble q-identities for mock theta functions or Maass waveforms, but the sign function prevents them from having the usual modular properties.

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  1. Agarwal, A.K., Andrews, G.E., Bressoud, D.M.: The Bailey lattice. J. Indian Math. Soc. 51, 57–73 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Andrews, G.E.: Multiple series Rogers–Ramanujan identities. Pacific J. Math. 114, 267–283 (1984)

    MathSciNet  Article  Google Scholar 

  3. Andrews, G.E.: The fifth and seventh order mock theta functions. Trans. Am. Math. Soc. 293, 113–134 (1986)

    MathSciNet  Article  Google Scholar 

  4. Andrews, G.E.: \(q\)-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Regional Conference Series in Mathematics, vol. 66. American Mathematical Society, Providence, RI (1986)

  5. Andrews, G.E.: Bailey’s Transform, Lemma, Chains and Tree, Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ). NATO Science Series II: Mathematics, Physics and Chemistry, pp. 1–22. Kluwer Acad. Publ., Dordrecht (2001)

    Book  Google Scholar 

  6. Andrews, G.E.: \(q\)-orthogonal polynomials, Rogers–Ramanujan identities, and mock theta functions. Proc. Steklov Inst. Math. 276(1), 21–32 (2012)

    MathSciNet  Article  Google Scholar 

  7. Andrews, G.E., Dyson, F.J., Hickerson, D.: Partitions and indefinite quadratic forms. Invent. Math. 91, 391–407 (1988)

    MathSciNet  Article  Google Scholar 

  8. Bressoud, D.M., Ismail, M., Stanton, D.: Change of base in Bailey pairs. Ramanujan J. 4, 435–453 (2000)

    MathSciNet  Article  Google Scholar 

  9. Bringmann, K., Kaszian, J., Milas, A., Nazaroglu, C.: Integral representations of rank two false theta functions and their modularity properties. Res. Math. Sci. 8, 54 (2021)

    MathSciNet  Article  Google Scholar 

  10. Bringmann, K., Lovejoy, J., Rolen, L.: On some special families of \(q\)-hypergeometric Maass forms. Int. Math. Res. Not. IMRN 18, 5537–5561 (2018)

    MathSciNet  Article  Google Scholar 

  11. Chen, R., Garvan, F.G.: A proof of the mod \(4\) unimodal sequence conjectures and related mock theta functions. Adv. Math. 398, 108235 (2022)

    MathSciNet  Article  Google Scholar 

  12. Cohen, H.: \(q\)-identities for Maass waveforms. Invent. Math. 91, 409–422 (1988)

    MathSciNet  Article  Google Scholar 

  13. Hikami, K.: Quantum invariant for torus link and modular forms. Commun. Math. Phys. 246(2), 403–426 (2004)

    MathSciNet  Article  Google Scholar 

  14. Hikami, K., Lovejoy, J.: Torus knots and quantum modular forms. Res. Math. Sci. 2, 2 (2015)

    MathSciNet  Article  Google Scholar 

  15. Hikami, K., Lovejoy, J.: Hecke-type formulas for families of unified Witten–Reshetikhin–Turaev invariants. Commun. Number Theory Phys. 11, 249–272 (2017)

    MathSciNet  Article  Google Scholar 

  16. Lovejoy, J.: A Bailey lattice. Proc. Am. Math. Soc. 132, 1507–1516 (2004)

    MathSciNet  Article  Google Scholar 

  17. Lovejoy, J.: Bailey pairs and indefinite quadratic forms. J. Math. Anal. Appl. 410, 1002–1013 (2014)

    MathSciNet  Article  Google Scholar 

  18. Mortenson, E.: On three third order mock theta functions and Hecke-type double sums. Ramanujan J. 30(2), 279–308 (2013)

    MathSciNet  Article  Google Scholar 

  19. Slater, L.J.: A new proof of Rogers’s transformations of infinite series. Proc. Lond. Math. Soc. 53(2), 460–475 (1951)

    MathSciNet  Article  Google Scholar 

  20. Warnaar, S.O.: 50 years of Bailey’s Lemma, Algebraic Combinatorics and Applications (Gößweinstein, 1999). Springer, Berlin (2001)

    Google Scholar 

  21. Zagier, D.: Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann). Astérisque 326, 143–164 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Zwegers, S.: Mock Theta Functions, PhD Thesis, Universiteit Utrecht (2002)

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Correspondence to Jeremy Lovejoy.

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Lovejoy, J. Bailey pairs and indefinite quadratic forms, II. False indefinite theta functions. Res. number theory 8, 24 (2022).

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  • Bailey pairs
  • False theta functions

Mathematics Subject Classification

  • 11F27
  • 33D15