Asymptotic formulae for mixed congruence stacks

  • Richard Frnka


Much like the important work of Hardy and Ramanujan (Proc Lond Math Soc 2(17):75–115, 1919) proving the asymptotic formula for the partition function, Auluck (Math Proc Camb Philos Soc 47:679–686, 1951) and Wright (Quart J Math (Oxf) 22:107–116, 1971) gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences on a two-parameter family of mixed congruence relations, with parts on one side up to the peak satisfying \(r \pmod {m}\) and parts on the other side \(-r\pmod {m}\). Techniques used in the proofs include Wright’s circle method, modular transformations, and bounding of complex integrals.


False theta functions Stacks Unimodal sequences Wright circle method 

Mathematics Subject Classification

05A15 05A16 05A17 11P81 11P82 



  1. 1.
    Abramowitz, M., Stegun, I.A.: Modified Spherical Bessel Functions. §10.2 In: Handbook of Mathematical Functions, 9th printing. Dover, New York, pp. 443–445 (1972)Google Scholar
  2. 2.
    Andrews, G.: An introduction to Ramanujan’s ‘lost’ notebook. In: Ramanujan: Essays and Surveys (AMS), vol. 22, pp. 165–184. Providence, RI (2001)Google Scholar
  3. 3.
    Andrews, G.: Concave and convex compositions. Ramanujan J. 31(1–2), 67–82 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Auluck, F.: On some new types of partitions associated with generalized Ferrers graphs. Math. Proc. Camb. Philos. Soc. 47, 679–686 (1951)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Beckwith, O., Mertens, M.H.: The number of parts in certain residue classes of integer partitions. Res. Number Theory 1(11), 1–15 (2015)MathSciNetMATHGoogle Scholar
  6. 6.
    Bringmann, K., Mahlburg, K.: Schur’s Second Partition Theorem and Mixed Mock Modular Forms. arXiv:1307.1800 [math.NT] (2013)
  7. 7.
    Davenport, H.: Multiplicative Number Theory, 2nd edn. Springer, New York (1980)CrossRefMATHGoogle Scholar
  8. 8.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 2(17), 75–115 (1919)MATHGoogle Scholar
  9. 9.
    Kim, B., Kim, E., Seo, J.: Asymptotics for q-expansions involving partial theta functions. Discrete Math. 338(2), 180–189 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Livingood, J.: A partition function with the prime modulus \(p >3\). Am. J. Math. 67, 194–208 (1945)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Meinardus, G.: Asymptotische aussagen über partitionen. Math. Z. 59, 388–398 (1954)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ngo, T.H., Rhoades, R.C.: Integer partitions, probabilities, and quantum modular forms. preprint,
  13. 13.
    Temperley, H.V.: Statistical mechanics and the partition of numbers II. The form of crystal surfaces. Proc. Camb. Philos. Soc. 48, 683–97 (1952)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis, p. 355. Cambridge University Press, Cambridge (1948)Google Scholar
  15. 15.
    Wright, E.M.: Stacks (II). Quart. J. Math. (Oxf) 22, 107–16 (1971)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wright, E.M.: Asymptotic partition formulae II: weighted partitions. Proc. Lond. Math. Soc. (92) 36, 117–141 (1933)MathSciNetMATHGoogle Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

Personalised recommendations