1,036 research outputs found
Partition theorems related to the Rogers-Ramanujan identities
AbstractIn this paper a partition theorem is proved which contains the Rogers-Ramanujan identities and Euler's partition theorem as special cases. Other partition theorems of the Rogers-Ramanujan type are proved
Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities
In this paper, we give a conjecture, which generalises Euler's partition
theorem involving odd parts and different parts for all moduli. We prove this
conjecture for two family partitions. We give -difference equations for the
related generating function if the moduli is three. We provide new companions
to Rogers-Ramanujan-Andrews-Gordon identities under this conjecture.Comment: 12 pages, revised versio
Partition theorems and the Chinese remainder theorem
The famous partition theorem of Euler states that partitions of into
distinct parts are equinumerous with partitions of into odd parts. Another
famous partition theorem due to MacMahon states that the number of partitions
of with all parts repeated at least once equals the number of partitions of
where all parts must be even or congruent to . These partition
theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and
Mugwangwavari. In this paper, we utilize the Chinese remainder theorem to prove
a comprehensive partition theorem that encompasses all existing partition
theorems. We also give a natural generalization of Euler's theorem based on a
special complete residue system. Furthermore, we establish interesting
congruence connections between the partition function and related
partition functions
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