Abstract
Let M(t, m, n) be the number of partitions of n with crank congruent to t modulo m. In this paper, we derive the generating functions of two variations of four combinations of the crank function. By considering the combinations of cranks of partitions modulo 12 and 16, we establish the generating functions of cranks:
where \(m=12,16\), and \(0\le t \le m/2\).
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This work was supported by the National Natural Science Foundation of China (No. 12201177), the Natural Science Foundation of Hebei Province (No. A2021205018), the Science and Technology Research Project of Colleges and Universities of Hebei Province (No. BJK2023092), the Doctor Foundation of Hebei Normal University (No. L2021B02), the Program for Foreign Experts of Hebei Province, and the Program for 100 Foreign Experts Plan of Hebei Province.
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Du, J.Q.D. On cranks of partitions modulo 12 and 16. Ramanujan J 62, 863–883 (2023). https://doi.org/10.1007/s11139-023-00718-0
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DOI: https://doi.org/10.1007/s11139-023-00718-0