Annals of Combinatorics

, Volume 23, Issue 3–4, pp 613–657 | Cite as

Finding Modular Functions for Ramanujan-Type Identities

  • William Y. C. ChenEmail author
  • Julia Q. D. Du
  • Jack C. D. Zhao


This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for \(a(mn+t)\). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for \(p(11n+6)\) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions \(\overline{p}(5n+2)\) and \(\overline{p}(5n+3)\) and Andrews–Paule’s broken 2-diamond partition functions \(\triangle _{2}(25n+14)\) and \(\triangle _{2}(25n+24)\). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews’ singular overpartition functions \(\overline{Q}_{3,1}(9n+3)\) and \( \overline{Q}_{3,1}(9n+6)\) due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.


Ramanujan-type identities Modular functions Generalized eta-functions Partition functions 

Mathematics Subject Classification

05A15 11P83 11P84 05A17 



We are grateful to Peter Paule for his inspiring lectures and for stimulating discussions. We would also like to thank the referees for their valuable comments and suggestions.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • William Y. C. Chen
    • 1
    Email author
  • Julia Q. D. Du
    • 1
  • Jack C. D. Zhao
    • 2
  1. 1.Center for Applied MathematicsTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Center for CombinatoricsNankai UniversityTianjinPeople’s Republic of China

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