Partition theorems related to some identities of Rogers and Watson

Author:
Willard G. Connor

Journal:
Trans. Amer. Math. Soc. **214** (1975), 95-111

MSC:
Primary 10A45

DOI:
https://doi.org/10.1090/S0002-9947-1975-0414480-9

MathSciNet review:
0414480

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper proves two general partition theorems and several special cases of each with both of the general theorems based on four *q*-series identities originally due to L. J. Rogers and G. N. Watson. One of the most interesting special cases proves that the number of partitions of an integer *n* into parts where even parts may not be repeated, and where odd parts occur only if an adjacent even part occurs is equal to the number of partitions of *n* into parts $\equiv \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7 \pmod 20$. The companion theorem proves that the number of partitions of an integer *n* into parts where even parts may not be repeated, where odd parts $> 1$ occur only if an adjacent even part occurs, and where 1â€™s occur arbitrarily is equal to the number of partitions of *n* into parts $\equiv \pm 1, \pm 2, \pm 5, \pm 6, \pm 8, \pm 9 \pmod 20$.

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Keywords:
Partitions,
<I>q</I>-series identities,
Rogers-Ramanujan identities,
generalized Rogers-Ramanujan identities,
frequency of parts,
Selbergâ€™s equation

Article copyright:
© Copyright 1975
American Mathematical Society