Abstract
An integral power series is called lacunary modulo M if almost all of its coefficients are divisible by M. Motivated by the parity problem for the partition function, Gordon and Ono studied the generating functions for t-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Okounkov, and Han.
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Acknowledgements
The authors would like to thank Ken Ono and Hannah Larson for advising this project, and for their many helpful conversations and suggestions.
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The authors were supported by Emory University, the Templeton World Charity Foundation, and the NSF via Grant DMS-1557690.
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Cotron, T., Michaelsen, A., Stamm, E. et al. Lacunary eta-quotients modulo powers of primes. Ramanujan J 53, 269–284 (2020). https://doi.org/10.1007/s11139-020-00257-y
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DOI: https://doi.org/10.1007/s11139-020-00257-y
Keywords
- Partitions
- Eta-quotients
- Nekrasov Okounkov formula
- Lacunary
- Eta-function
- Modular forms
- Generalized eta-quotients
- Powers of primes