Series and polynomial representations for weighted Rogers-Ramanujan partitions and products modulo 6.(English)Zbl 1151.11348

Akiyama, Shigeki (ed.) et al., Probability and number theory – Kanazawa 2005. Proceedings of the international conference on probability and number theory, Kanazawa, Japan, June 20–24, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-43-3/hbk). Adv. Stud. Pure Math. 49, 21-39 (2007).
Summary: We obtain infinite series representations for certain weighted Rogers-Ramanujan partitions which we recently showed are related to partitions into parts $$\not\equiv 0,\pm i\pmod 6$$, for $$i=1,2,3$$. We also show that our series can be transformed into the series previously obtained by Bressoud which connect the partitions into parts $$\not\equiv 0,\pm i\pmod 6$$ with partitions satisfying certain bounds on their successive ranks. Finally, we obtain finite versions of our series representations, namely, polynomial identities which tend to the infinite series identities when certain parameters tend to infinity.
For the entire collection see [Zbl 1132.11001].

MSC:

 11P83 Partitions; congruences and congruential restrictions 05A19 Combinatorial identities, bijective combinatorics