Ramanujan congruences for overpartitions with restricted odd differences. (English) Zbl 1515.11101
S. Ahlgren and M. Boylan [Invent. Math. 153, No. 3, 487–502 (2003; Zbl 1038.11067)] showed that the Ramanujan congruences \(p(ln+a)\equiv 0\pmod l, \, l=5,7,11\) are the only ones for \(p(n)\). In the present article, the authors generalize the work of J. Sinick [Int. J. Number Theory 6, No. 4, 835–847 (2010; Zbl 1205.11112)], and investigate Ramanujan congruences for the function \(\overline{t}(n)\), which counts the overpartitions of \(n\) with restricted odd differences, and as a particular case they show that only one such congruence exists. The authors also provide two congruences modulo \(5\) for \(\overline{t}(n)\), namely \(\overline{t}(80n+40)\equiv 0\pmod 5\) and \(\overline{t}(80n+60)\equiv 0\pmod 5\). The authors also generalize Theorem \(1.2\) of Sinick [loc. cit.] to bound Ramanujan congruences for more general eta-quotients. All the results presented in the article are explained properly with supporting arguments.
Reviewer: M. P. Chaudhary (New Delhi)
MSC:
| 11P83 | Partitions; congruences and congruential restrictions |
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
Software:
eta-quotientsReferences:
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