Finding Modular Functions for Ramanujan-Type Identities

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 ( m n + t ) . W...

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Bibliographic Details
Published in:Annals of combinatorics 2019-11, Vol.23 (3-4), p.613-657
Main Authors: Chen, William Y. C., Du, Julia Q. D., Zhao, Jack C. D.
Format: Article
Language:English
Subjects:
Algorithms
Combinatorics
Diamonds
Dissection
Identities
Mathematics
Mathematics and Statistics
Partitions (mathematics)
Quotients
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Summary: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 ( m n + t ) . While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p ( 11 n + 6 ) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions p ¯ ( 5 n + 2 ) and p ¯ ( 5 n + 3 ) and Andrews–Paule’s broken 2-diamond partition functions ▵ 2 ( 25 n + 14 ) and ▵ 2 ( 25 n + 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 Q ¯ 3 , 1 ( 9 n + 3 ) and Q ¯ 3 , 1 ( 9 n + 6 ) due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.
ISSN:0218-0006
0219-3094
DOI:10.1007/s00026-019-00457-4