Some new identities for colored partitions with parts in multiples of $ 4 $

Roberta R. Zhou; Fuquan Ren; Roberta R. Zhou; Fuquan Ren

doi:10.3934/math.20241300

AIMS Mathematics

2024, Volume 9, Issue 10: 26721-26740. doi: 10.3934/math.20241300

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Some new identities for colored partitions with parts in multiples of $ 4 $

Roberta R. Zhou ^{1
,
,},
Fuquan Ren ²

1.
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Hebei 066004, China
2.
School of Science, Yanshan University, Hebei 066004, China

Received: 11 July 2024 Revised: 03 September 2024 Accepted: 05 September 2024 Published: 13 September 2024
MSC : Primary 11P84; Secondary 05A15, 05A17

To start, we provided the deformations of certain specific modular equations derived from Ramanujan's notebook, Part Ⅲ, by using the complete square formula. Subsequently, we established fifteen identities for partitions with distinct colors that arose from these deformed modular equations in the theory of modular equations. Throughout our investigation, we found that the fifteen identities have parts that were in multiples of $ 4 $.
- colored partitions,
- modular equations,
- theta function identities
Citation: Roberta R. Zhou, Fuquan Ren. Some new identities for colored partitions with parts in multiples of $ 4 $[J]. AIMS Mathematics, 2024, 9(10): 26721-26740. doi: 10.3934/math.20241300

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Abstract

To start, we provided the deformations of certain specific modular equations derived from Ramanujan's notebook, Part Ⅲ, by using the complete square formula. Subsequently, we established fifteen identities for partitions with distinct colors that arose from these deformed modular equations in the theory of modular equations. Throughout our investigation, we found that the fifteen identities have parts that were in multiples of $ 4 $.

References

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