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Finding Modular Functions for Ramanujan-Type Identities

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    William Y. C. Chen & Julia Q. D. Du

  2. Center for Combinatorics, Nankai University, Tianjin, 300071, People’s Republic of China

    Jack C. D. Zhao

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  1. William Y. C. Chen
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Correspondence to William Y. C. Chen.

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Dedicated to Professor George E. Andrews on the occasion of his 80th birthday

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Chen, W.Y.C., Du, J.Q.D. & Zhao, J.C.D. Finding Modular Functions for Ramanujan-Type Identities. Ann. Comb. 23, 613–657 (2019). https://doi.org/10.1007/s00026-019-00457-4

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